Asymptotic analysis of solutions related to the game-theoretic p-laplacian
Diego Berti

1.1

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    < g r a p h i c s >

	 
	  Università di Firenze, Università di Perugia, INdAM  consorziate nel CIAFM

	 
	 

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	DOTTORATO DI RICERCA

	IN MATEMATICA, INFORMATICA, STATISTICA

		 .2cm
		 CURRICULUM IN MATEMATICA

	CICLO XXXI
	

	
		 
	 Sede amministrativa Università degli Studi di Firenze

	Coordinatore Prof. Graziano Gentili
	

	Asymptotic analysis of solutions
related to the 
 game-theoretic p-laplacian
	

	
	Settore Scientifico Disciplinare MAT/05





0.42
	Dottorando:
	

	Diego Berti


0.42
	Tutore
	

	Prof. Rolando Magnanini
	






	0.30
		Coordinatore
		

		Prof. Graziano Gentili
	






	

	
	Anni 2015/2018
	
















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[LE]
[RO]
[RE,LO]






















CHAPTER: ACKNOWLEDGMENTS

Here, I mention those who have foremost contributed to the writing of this work as
the conclusion of my Ph.D.
First of all, I express my gratitude to my advisor Prof. Rolando Magnanini for his constant
 help in the production of this thesis and in the progress of my academic figure.
He first introduced me to the topics and questions I have studied in this thesis and his
competence, passion and experience have been crucial in more than one step.
I want to thank the members of the research group in Firenze, for involving me and
for their precious support through these years. In particular, a quote is due to Prof.
Paolo Salani, Prof. Chiara Bianchini, Prof. Andrea Colesanti and Prof. Elisa Francini.
Also, for having shared most of the steps of the journey with good advice, I thank my
academic brother Dr. Giorgio Poggesi.


The visits to the prestigious Research Center for Pure and Applied Mathematics
 at the Tohoku University in Sendai have been 
extremely significant for all my work. I truly thank Prof. Shigeru Sakaguchi from
his warm hospitality. Exchanges of ideas with him and his research group have been
beneficial. I have appreciated the possibility of establishing contact with some young
researchers from the other side of the world.
I would like to thank Prof. Mikko Parviainen for the hint that improved Lemma
1.23. I met him and his research group during a refreshing and stimulant week of
 the Summer School in Jyväskylä.
I wish to thank the doctorate program in Firenze and the Dipartimento di Matematica
e Informatica “Ulisse Dini” in Firenze. In particular, I mention the director of
my Ph.D. program Prof. Graziano Gentili, for his attention to the needs of doctoral
students. I also thank my Ph.D. program for its essential financial support.
For their priceless understanding, comfort and motivation, I would like to remark
the importance of my companions in the doctoral students room T/8, especially who
 I shared most of the last three years.
Finally, I wish to end these acknowledgments dedicating special thanks to the support
I received during my life by my family and during the last two years by Masoomeh.
Thank you.































CHAPTER: INTRODUCTION

tocchapterIntroduction






The game-theoretic (or normalized or one-homogeneous) p-laplacian is defined as

    Δ_p^G u = 1/p|∇ u|^2-p_p u,

for p ∈ (1,∞), where _p u is the usual p-laplacian 

    _p u = div(|∇ u|^p-2∇ u),

and as

    Δ_∞^G u = ⟨∇^2 u ∇ u, ∇ u ⟩/|∇ u|^2,

in the extremal case p=∞. With these notations, by computing formally the divergence in (<ref>), _p^G can be written for any p∈ [1,∞] as

    _p ^G u = 1/p u +(1 -2/p) _∞ ^G u,

where  is the classical Laplacian.



Works by Peres, Schramm, Scheffield and Wilson (see <cit.>) have emphasized the role of _p^G in stochastic differential equations in the context of game theory. Indeed, equations for _p^G appear when one considers the limiting value for vanishing length of steps of certain two-players games called tug-of-war (or TOW) games with noise in the case p∈ (1,∞) and TOW games in the case p = ∞. In this context, it is possible to consider a stochastic game between two opponents, one who wants to maximize and the other who wants to minimize the payoff. Heuristically, the game consists of a combination of random moves (which correspond to noise and are dictated by ) and moves that are orthogonal to the gradient (which correspond to TOW and depend on the operator _∞^G). In this context, we also mention works by Manfredi, Parviainen and Rossi, in <cit.>.
The relation between TOW games and differential equations similar to those involving the game-theoretic p-laplacian are considered by Nyström and Parviainen in the context of market manipulation and option pricing (see <cit.>).
There is also a growing interest for equations involving the game-theoretic p-laplacian in connection to numerical methods for image enhancement or restoration (see <cit.> and <cit.>). Typically, for a possibly corrupted image represented by a function u_0, it is considered an evolutionary process based on _p^G with initial data u_0 and homogeneous Neumann boundary conditions. As explained in <cit.>, the different choice of p affects the direction in which the brightness evolves; the 1-homogeneity of _p^G ensures that such an evolution does not depend on the brightness of the image. The relation between solutions of parabolic equations and a corresponding parametrized elliptic equation is examined in <cit.> for the classical p-laplacian, and can be extended to the case of _p^G in hand.

Besides the cited applications, problems for _p^G have been recently studied by Attouchi and Parviainen <cit.>, Attouchi, Parviainen and Ruosteenoja <cit.>, Parviainen and Ruosteenoja <cit.>, A. Björn, J. Björn and Parviainen <cit.>, Parviainen and Vázquez <cit.>, Banerjee and Garofalo <cit.>, Does <cit.>, Juutinen and Kawohl <cit.>, Kawohl, Krömer and Kurtz <cit.> as well as Banerjee and Kawohl <cit.> and Jin and Silvestre <cit.>.






Observe that, having in mind the formal decomposition (<ref>), when p=2 we simply obtain _2^G =  / 2 and that, for p ≠ 2, _p^G can be seen as a proper singular perturbation of / p. Indeed, one notices that _∞^G u has discontinuous coefficients when u = 0.
Nevertheless, _p^G is uniformly elliptic (in case p∈ (1,∞)) and (degenerate) elliptic in the case p=∞.
It is evident that, for p≠2, ^G_p is nonlinear. However, _p^G is somewhat reminiscent of the lost linearity of the Laplace operator, since it is 1-homogeneous, that is 

    _p^G (  u ) = _p^G u   ∈,

 differently from _p, which instead is (p-1)-homogeneous. The nonlinearity of  _p^G is indeed due to its non-additivity. Nevertheless, _p^G acts additively if one of the relevant summands is constant and, more importantly, on functions of one variable and on radially symmetric functions. We shall see in the sequel that these last properties are decisive for the purposes of this thesis.
Also, when p ≠ 2, differently from _p, _p^G is not in divergence form. This fact implies that we cannot apply the standard theory of distributional weak solutions. We need to consider the theory of viscosity solutions. The main tool of this theory we will use is the comparison principle, that we recall and adapt to our purposes in Chapter <ref>, together with some versions of the strong maximum principle and the Hopf-Oleinik lemma, which will be used in Chapter <ref>.




In this thesis, we focus on the connection between asymptotic formulas for solutions of certain game-theoretic p-laplacian problems and some geometrical features of the relevant domain. 
We will generally consider a domain  in ^N, with N≥ 2, not necessarily bounded, with boundary ≠∅.
We shall consider viscosity solutions u=u(x,t) of the following initial-boundary value problem:

    u_t=_p^G u      ×(0,∞), 
       u=0       ×{ 0}, 
       u=1       ×(0,∞).

Also, we shall consider viscosity solutions u^ of the one-parameter family of boundary value problems

    u^-^2_p^G u^=0      , 
       u^=1       .

 Our attention will focus on the asymptotic analysis for small positive values of parameters t and .
 
 
 
 
When p=2, the two cases are strongly connected. Indeed, by taking advantage of the linearity of (<ref>), one can use a modified Laplace transform, to obtain that, 

    u^(x) = 1/^2∫_0^∞ u(x,t) e^-t/^2 dt   x ∈,  > 0.

In the case p≠ 2 the two problems are no more equivalent. Nevertheless, if  has spherical or one-dimensional symmetry, due to the fact that for radial or one-dimensional functions the operator _p^G acts linearly, (<ref>) still holds true. This observation will be crucial. A proper manipulation of (<ref>) will give suitable barriers to estimate the parabolic solution (see Lemma <ref>).



In what follows, we will describe the main results of this thesis, which are mainly contained in the two papers <cit.>. The case of problem (<ref>)-(<ref>) is considered in <cit.>, whereas <cit.> addresses the case of problem (<ref>)-(<ref>).
Under the assumption that  merely satisfies the topological assumption = (^N ∖),  in  Theorem <ref> we establish for p∈ (1,∞] the asymptotic profile of the solution of (<ref>)-(<ref>) for small values of time:

    lim_t → 0^+ 4 t log u(x,t) = -p'  d_(x)^2,  x∈.

 Here, by p' we mean the conjugate exponent of p, that is 1/p + 1/p' = 1, for p∈(1,∞), and p'=1, when p=∞. Also, by d_(x), we mean the distance of the point x ∈ from the boundary , defined by

    d_(x) = inf{|x- y|: y ∈},  x∈.

Moreover, in Theorem <ref>, we obtain the corresponding formula for the solution of the elliptic problem (<ref>)-(<ref>):

    lim_→ 0^+log u^(x) = -√(p')  d_(x),  x∈.

These pointwise asymptotic profiles extend known formulas in the linear case, first obtained by Varadhan by using analytic methods (see <cit.>). See also <cit.>, where Evans and Ishii used arguments pertaining the theory of viscosity solutions, and <cit.>, for a treatment with probabilistic methods by Freidlin and Wentzell. These formulas are common in the context of large deviations theory. There, random differential
 equations with small noise intensities are considered. The profiles for small values of  and t of the solutions of (<ref>)-(<ref>) and (<ref>)-(<ref>) are respectively related to the behavior of the exit time and to the probability to exit from  of a certain stochastic process (see <cit.> and <cit.>).  
 Formulas (<ref>) and (<ref>) are obtained by employing barrier arguments based on accurate estimates on radial solutions. In particular, all we need is to control solutions both in a ball or in the complement of a ball. In the elliptic case we are able to compute them in terms of Bessel functions whereas the parabolic case is more delicate, since we need to properly involve (<ref>) and the existence of a global solution of (<ref>).
We mention that proper versions of formula (<ref>) are included in a series of works by Magnanini-Sakaguchi in linear cases (see <cit.>), in certain nonlinear contexts (see <cit.>) and concerning initial-value problems  (<cit.>). 







A second type of asymptotic formulas that we obtain, which deeply link solutions of  (<ref>)-(<ref>) or (<ref>)-(<ref>) to the geometry of the domain, involve certain statistical quantities called q-means. Given x∈, t>0 and q∈ (1,∞], the q-mean μ_q(x,t) on B_R(x)⊂ of the solution u of (<ref>)-(<ref>) is the unique real number μ such that
 
    ‖ u (·,t)- μ‖_L^q(B_R(x))≤‖ u(·,t) -‖_L^q(B_R(x))  ∈.

 Observe that μ_q generalizes the mean value of u on B_R(x), which is obtained when one chooses q=2 in the above definition.
 Consider a domain of class C^2. Suppose  that there exists y_x ∈ such that (^N∖)∩ B_R(x)={ y_x}, where R= d_(x). Assume that _1(y_x),…,_N-1(y_x) < 1/R, where we have denoted with _1,…,_N-1 the principal curvatures with respect to the inward normal of  at y_x, and set

    Π_(y_x) =∏_j=1^N-1[1-R _j(y_x)].

In Theorem <ref> we establish that, for any q ∈(1,∞) and p∈ (1,∞],



















 
    lim_t→ 0^+(R^2/t)^N+1/4(q-1)μ_q(x,t)= C_N,p,q{Π_Γ(y_x)}^-1/2(q-1),

where C_N,q,p is a positive constant only depending on the labelled parameters.






In the extremal case q=∞ and for any p∈ (1,∞], we obtain that

    μ_∞(x,t) →1/2  t → 0^+.

Analogously, from an accurate improvement of barriers in the case of smooth domains we compute the asymptotic profile of μ_q,, the q-mean of u^ on B_R(x). In fact, in Theorem <ref> we show that

    lim_→ 0^+(R/)^N+1/2(q-1)μ_q,(x)=
      C_N,p,q{Π_(y_x)}^-1/2(q-1).








Also in this case, we obtain that

    μ_∞,(x) →1/2  → 0^+.
















We emphasize that (<ref>) and (<ref>) generalize, to each p∈(1,∞], and extend, to any q ∈(1,∞), known formulas in the linear case, for p = q = 2. In fact, we recall that in <cit.>, it has been given the following asymptotic formula for the so-called heat content on B_R(x):

    lim_t→ 0^+
    t^-N+1/4∫_B_R(x)u(z,t) dz=
    C_N R^N-1/2/√(Π_Γ(y_x)),

where u is the solution of the heat equation satisfying (<ref>) and (<ref>). A normalization makes apparent the connection of the heat content to μ_2(x,t). See also  <cit.> for a similar formula in the elliptic case. Similar formulas for the case q=2 in nonlinear settings can be found in <cit.> for the evolutionary p-Laplace equation, and in <cit.>, for a class of non-degenerate fast diffusion equations. 
It is worth noting that (<ref>) and (<ref>) are essentially based on two ingredients. The first is the geometrical lemma <cit.> which determines the behavior of _N-1(_s ∩ B_R(x)) for vanishing s > 0 in terms of the function Π_. Here, _s = { x∈: d_(x) =s }, for s > 0 and _N-1 is the (N-1)-Hausdorff measure. The second ingredient is the construction of 
 
 
 
 sharp uniform estimates of Varadhan-type formulas (<ref>) and (<ref>), which are novelty even in the case p=2.
In the case  is of class C^0,, that is  is locally a graph of a continuous function with modulus of continuity controlled by  (see Section <ref>), we provide in Theorem <ref> the following estimate:
 
    4  t log u(x,t) + p'  d_(x)^2 = O(t  logψ_(t)),

 for t→ 0^+, uniformly on every compact subset of . Here, ψ_(t) is a function that depends on  and is positive and vanishes as t → 0^+. In particular, if  is smooth enough, (<ref>) gives a sharp estimate of the rate of convergence in (<ref>). In fact, if for example  is a -Hölder domain, for some 0 <  < 1, we obtain that the right-hand side of (<ref>) is O(t log t), as t→ 0^+.
 In the elliptic case, explicit barriers are available, 
we obtain more accurate uniform estimates. In fact, in Theorem <ref>, we prove that 
 
    log u^(x) + √(p') d_(x) = 
      
      O()   p=∞,
     
      O( log)   p ∈ (N, ∞),

  as → 0^+, on every compact subset of .  In the case of a domain  of class C^0,, it holds instead that
  
    log u^(x) +√(p')  d_(x)=
       
      O( log |logψ_()|)  p=N,
      
    
      O(logψ_())   p∈(1,N),

  for → 0^+, uniformly on every compact subset of .
  We observe that the presence of the threshold for the exponent p in this last formula seems to be connected to the integrability of the global solution of (<ref>) with respect to the variable t ∈ (0,∞). This suggests that even in the parabolic case we may expect this kind of behavior. We were not able to prove it so far.
  Finally, notice that, by using comparison results, formulas (<ref>) and (<ref>)-(<ref>) can be easily extended to the case of a prescribed non-constant data on the boundary. See Corollaries <ref> and <ref>.





In Chapter <ref>, the obtained Varadhan-type formulas and formulas for q-means will find applications to geometric and symmetry results. The linearity of   was used in <cit.> to derive radial symmetry of compact stationary isothermic surfaces, that is those level surfaces of the temperature which are invariant in time. In Chapter <ref>, we will extend this type of result to the case p 2. In the case p=2, it was shown that the mean values μ_2(x,t) or   μ_2,(x) do not depend on x if this lies on a stationary isothermic surface, and hence, for instance, (<ref>) gives that

    ∋ y ↦Π_(y)
 
The radial symmetry then ensues from Alexandrov's Soap Bubble Theorem for Weingarten surfaces (see <cit.>). 
For p≠2, this approach is no longer possible. However, when  is compact, an approach based on the method of moving planes (see <cit.>, as in <cit.> and <cit.>) is still feasible. We also treat a case in which  is unbounded, by using the sliding method (see <cit.>), as in <cit.>, <cit.>, <cit.> and <cit.> to obtain that  must be a hyperplane. We stress that a crucial step to apply the cited methods in our cases is the application of the (classical) strong comparison principle in a suitable subset (as done in <cit.> or in <cit.>), which is determined by an application of the strong maximum principle and Hopf lemma for viscosity solutions.







We conclude this introduction by a summary of this thesis.
Chapter <ref> recalls those technical tools of the theory of viscosity solutions which we will use in the remaining chapters. 
In Chapter <ref>, we consider the cases of symmetric domains, in which _p^G acts as a linear operator. This allows us to deal with explicit solutions of (<ref>)-(<ref>) and (<ref>)-(<ref>) and to compute their asymptotic profiles. The explicit formulas give sharp estimates that will be crucial to control the case of generic domains.
Formulas of Varadhan-type, that is (<ref>) and (<ref>), are collected in Chapter <ref>. There, also their uniform sharp versions  (<ref>) and (<ref>)-(<ref>) shall be given.
In Chapter <ref>, we provide our asymptotic formulas for q-means  (<ref>) and (<ref>).
Finally, Chapter <ref> contains a few applications of the formulas derived in Chapters <ref> and <ref>. In particular, we provide generalization of symmetry results present in the literature.











CHAPTER: PRELIMINARIES ON THE THEORY OF	 VISCOSITY SOLUTIONS


 
 Several results of this thesis are based on some important properties of viscosity solutions. In the next chapters  we shall use comparison principles and we shall apply strong maximum principles and the Hopf-Oleinik Lemma. Since _p^G has discontinuous coefficients, standard results for viscosity solutions cannot be directly applied, but they must be adapted. In this chapter we describe how, by pointing out the significant references.
 
 A recent summary on aspects of viscosity solutions of our interest, which includes a quite complete list of references, is a  dedicated chapter in <cit.> (where the theory is instrumental to the study of surface evolution equations). We adopt that approach. Useful references on definitions and relevant properties of viscosity solutions are also the classical surveys <cit.>, besides the beginner's guide <cit.>. For more recent and specific works on _p^G, where viscosity solutions are adopted, we give the following (not complete) list of publications: <cit.>.

In Section <ref>, we shall begin with definitions and relevant properties of viscosity solutions of general singular differential equations. In this more general context, we shall also state those theorems (from <cit.>) which we shall apply to our cases, in Section <ref>. 
In Section <ref>, in the specific case of the game-theoretic p-laplacian, we shall collect those results which we will use in the next chapters of this thesis. In particular, in Subsection <ref>, for both equations (<ref>) and (<ref>), we will give comparison principles (see Corollaries <ref> and <ref>) as well as corresponding strong maximum principles (see Corollaries <ref> and <ref>).

Finally, in Subsection <ref> we shall extend a sharp version of Hopf-Oleinik lemma (obtained by Mazya et al., in <cit.>), to viscosity solutions of (<ref>)-(<ref>) and (<ref>)-(<ref>) (see Corollaries <ref> and <ref>). 








§ VISCOSITY SOLUTIONS OF SINGULAR DIFFERENTIAL EQUATIONS



To start with, we introduce some definitions on arbitrary functions, which we use in this chapter.  Let f be a function on a metric space , with values in . We recall that a function f is called lower semicontinuous if

    lim inf_y→ x f(y) ≥ f(x)

and that a function f is called upper semicontinuous if -f is lower semicontinuous. 
For a function f, the lower semicontinuous envelope f_* is defined by 

    f_*(z)= lim_→ 0^+inf{f(z'): z'∈ B_(z)⊂},

and the upper semicontinuous envelope f^* is defined by f^*=-(-f)_*. Note that the previous definition differs from that of lim inf, since the infimum is taken upon the whole ball B_(z). Of course, if f is continuous, f and its envelopes coincide. 















 For N≥ 1, let ^N be the linear space of N× N symmetric matrices. An operator F:×(0,∞)××(^N∖{0})×^N →, is called (degenerate) elliptic if it satisfies

    F(x,t,u,ξ,X) ≥ F(x,t,u,ξ, Y)   X≤ Y,

for any (x,t,u,ξ)∈×(0,∞)××(^N∖{0}) and X,Y ∈^N.
Here with X≤ Y we mean that (X-Y) η, η≤ 0, for any η∈^N.
We give the following definitions (see <cit.>). A lower semicontinuous function, u:×(0,∞)→,  is a viscosity subsolution of 

    u_t+ F(x, t, u,  u, ^2 u) = 0   ,

if for any (x,t,ϕ)∈×(0,∞)× C^2(× (0,∞)), such that u-ϕ attains its maximum at (x,t), it holds that

    ϕ_t(x,t)+
    F_* (x, t, u(x,t), ϕ(x,t), ^2 ϕ (x,t)) ≤ 0.

Analogously, an upper semicontinuous function, v:×(0,∞)→, is called a viscosity supersolution of (<ref>), if for any (x,t,ψ)∈×(0,∞)× C^2(× (0,∞)) such that v-ψ attains its minimum at (x,t), it holds that

    ψ_t(x,t)+
    F^* (x, t, v(x,t), ψ(x,t), ^2 ψ (x,t)) ≥ 0.

Finally, we say that a function u is a viscosity solution of (<ref>) if it is both a viscosity subsolution and a viscosity supersolution. 











The next lemma affirms that the theory is consistent with the classical definition of solutions. The proof is straightforward.



Assume that F is (degenerate) elliptic.
Let u∈ C^2(×(0,∞)) be such that

    F_*(x,t, u,  u, ^2 u) ≤ u_t ≤ F^*(x,t, u,  u, ^2 u),

for every (x,t)∈× (0,∞), then u is a viscosity solution of (<ref>).


For our purposes, we will need the following extension lemma, that claims  that, for smooth functions, it suffices to check the definitions away from isolated critical points.



Assume that F is (degenerate) elliptic. 
Let u∈ C^2(×(0,∞)) such that


  * (x_0 , t) ∈× (0,∞) is the unique point in × (0,∞) such that

    u (x_0,t) = 0.


  * In ( ∖{x_0}) × (0,∞), it holds that
 
    F_*(x,t, u,  u, ^2 u) ≤ u_t ≤ F^*(x,t, u,  u, ^2 u).


Then u is a solution of (<ref>) in × (0,∞).



Take a sequence of points y_n∈∖{x_0} such that y_n→ x_0 as n→∞.  From our assumption on u, we have that both of the following inequalities hold at (y_n,t):

    u_t + F_*(x, t, u,  u, ^2 u ) ≤ 0,

and

    u_t + F^*(x, t, u,  u, ^2 u )≥ 0.

Now, since F_* is lower semicontinuous and F^* is upper semicontinuous we have that

    u_t ( x_0 , t ) +F_*(x_0, t, u(x_0),  u(x_0), ^2 u(x_0) ) ≤
    lim inf_n→∞{u_t ( y_n , t ) +F_*(y_n, t, u(y_n),  u(y_n), ^2 u(y_n) )}≤ 0

and 

    u_t ( x_0 , t ) +F^*(x_0, t, u(x_0),  u(x_0), ^2 u(x_0) ) ≥
    lim sup_n→∞{ u_t ( y_n , t ) +F^*(y_n, t, u(y_n),  u(y_n), ^2 u(y_n) )}≥ 0.

The claim follows, thanks to Lemma <ref>.




Obvious adjustments of definitions and Lemmas <ref> and <ref> are given in the case of singular elliptic differential equations (see for example <cit.>).
 






















 §.§ Comparison principles



For our purposes, we propose quite general comparison results, in a not necessarily bounded domain . The relevant assumptions apply to the differential equations (<ref>) and (<ref>).
As a modulus of continuity we mean a continuous function :[0,∞)→ [0,∞), such that (0)=0. If X ∈^N, by |X| we intend the operator norm of X on ^N.



Assume that F:^N∖{0}×^N→ is continuous and (degenerate) elliptic. In addition, let F satisfy the following properties:
 

  * it holds that

    -∞ < F_*(0,0) = F^*(0,0) < +∞;



  * for every R>0,

    sup{|F(ξ, X)|: 0< |ξ|≤ R,  |X|≤ R} < ∞.


Let u and v be, respectively, a subsolution and a supersolution of 

    u_t + F( u, ^2 u)=0

 in × (0,∞). Assume that


  * u(x,t) ≤ K(|x|+1), v(x,t) ≥ -K (|x|+1), for some K>0  independent of (x,t)∈× (0,∞);

  * there is a modulus  such that 

    u^*(x,t) - v_*(y,t) ≤(|x-y|),

for all  (x,y,t)∈(×)×(0,∞)∪(×)×{0};

  * u^*(x,t) - v_*(y,t) ≤ K(|x-y| + 1) on (×)×(0,∞)∪(×)×{0}, for some K>0 independent of (x,y,t).

Then, it holds that 

    u^*≤ v_*   × (0,∞).

 



Note that (ii) is equivalent to the condition u^* ≤ v_* on ×(0,∞) and that (i) and (iii) are unnecessary, if  is bounded. 
It is also evident that, if u^* and v_* are bounded, then (i) and (iii) are satisfied.



We now state a general comparison principle for elliptic equations, which is a corollary of <cit.>, that treats general equations.



Let F:^N∖{0}×^N→ be continuous, (degenerate) elliptic and such that:


  * 
    -∞ < F_*(0,0) = F^*(0,0) < +∞;



  * for every R>0,

    sup{|F(ξ, X)|: 0< |ξ|≤ R, |X|≤ R } < ∞;


  * for every R>ρ>0, there exists a modulus  such that

    |F(x,r,ξ,X)-F(x,r, η, X)| ≤(|ξ-η|),

for all x∈, r∈, ρ≤ |ξ|, |η|≤ R, |X|≤ R.


Let u and v be, respectively, a subsolution and a supersolution of 

    u + F(  u, ^2 u) =0   .

Moreover, suppose that


  * u(x) ≤ K(|x|+1), v(x) ≥ -K (|x|+1), for some K>0  independent of x∈;

  * there is a modulus  such that 

    u^*(x) - v_*(y) ≤(|x-y|),   (x,y)∈(×);


  * u^*(x) - v_*(y) ≤ K(|x-y| + 1) on (×), for some K>0 independent of (x,y)∈(×).

Then, it holds that

    u^*≤ v_*  .





Observe that assumptions (a) and (b) of Theorem <ref> are fulfilled, if both F_* and F^* are continuous in their variables. In particular, the supremum in (b) is less than or equal to

    sup{|F_*(ξ,X)| + |F^*(ξ,X)|: |ξ|, |X| ≤ R}.






 §.§ Strong maximum principles



In Chapter <ref> we shall use the strong maximum principle for (<ref>) and (<ref>).  Here, we state results for a quite large class of differential operators. 
We start with the next theorem, which can be seen as a corollary of <cit.>.

 
 
 Assume that F:^N∖{0}×^N→ is lower semicontinuous and (degenerate) elliptic. Moreover, assume that F satisfies the following requirements:
 




 
  * there exists ρ_0 > 0 such that , for any choice of 0 < |s|, |ξ| <ρ_0, there exists _0≥ 0 such that
 
    s + F(ξ, I-ξ⊗ξ) >0

 holds for every ≥_0>0;
 
  * for all η >0 there exist a function φ :(0,1) → (0,∞), _η>0 and _0 ≥ 0 such that for all ∈(0, _η] and > _0,
 
    s + F_*(ξ, (I - ξ⊗ξ)) ≥φ () [s+F_*(ξ,( I -ξ⊗ξ))]

 holds for all  0< |ξ|≤η, |s|≤η.
 
  * there exists _0>0 such that	
 
    1+F_*(0,  I) >0

 for all 0<  <_0.
 
  * for all η >0 there exist φ:(0,1) → (0,∞), _η>0 such that for each K>0 and for each ∈ (0,_η], then
 
    s + F_*(2 K  (y-x), 2K  I) ≥
    φ () [s + F_*(2K (y-x), 2K I)],

 holds for all -η≤ s≤ 0.
 
 Let u be a viscosity subsolution of (<ref>)



 in × (0,∞).  Suppose that u achieves a  maximum at (x_0,t_0)∈× (0,∞). 
 Then u is constant on the set of all points, which can be
connected to (x_0,t_0) by a simple continuous curve in ×(0,t_0) along which the t-coordinate is nondecreasing.
 
 
 
  
  
Note that a strong minimum principle also holds for (<ref>), once the obvious necessary changes in Theorem <ref> are made.
 
 


We now give the elliptic version of the strong maximum principle, which will be applied to solutions of equation (<ref>). We report a corollary of	<cit.>, for a general operator G=G(u,  u, ^2 u ). Then, in next section, we shall apply to the case of the game-theoretic p-laplacian.


Let  be a connected set, G:×^N∖{0}×^N →.  Assume that G is lower semicontinuous and satisfies
[(a)]
 
  * for any  s,r ∈, ξ∈^N∖{0} and X,Y∈^N, it holds that
  
    G (r,ξ,X)≤ G(s,ξ, Y)   s≥ r   Y≤ X;

 
  * for every η > 0, there exists a function φ: (0,1) → (0,1) such that 
 
    G(  r, ξ,  X ) ≥φ() G( r, ξ, X )

 holds for all r∈[-1,0], 0 < |ξ| ≤η, |X|≤η;
 
  * there exists ρ_0 > 0 such that for any choice of 0 < |ξ| ≤ρ_0, 
 
    G( 0, ξ, I - ξ⊗ξ) > 0    > _0

 holds for some _0≥ 0.

Suppose that u be a viscosity subsolution of 

    G(u,  u, ^2 u ) = 0




in  that achieves a nonnegative maximum in . Then u is constant in .




§ THE CASE OF THE GAME-THEORETIC P-LAPLACIAN



Now, we consider the case of the game-theoretic p-laplacian, that is the differential equations (<ref>) and (<ref>). To begin with, we observe that _p^G can be formally seen as a singular, quasi-linear operator F: (^N∖{0})×^N →, where

    F(ξ,X)=-[A(ξ) X],

with

    A(ξ)=1/p I+ (1-2/p) ξ⊗ξ/|ξ|^2

for ξ∈^N∖{0}.
Here, I denotes the N× N identity matrix.
Observe that, if ξ≠ 0, F=F_*=F^* while, if ξ=0, we can explicitly calculate the semicontinuous envelopes of F (see <cit.> or <cit.>). It holds that

    p F_*(0,X)=-(X)-min(p-2,0) (X)-max(p-2,0) (X),
    
    p F^*(0,X)=-(X)-max(p-2,0) (X)-min(p-2,0) (X),

where (X) and (X) are the maximum and minimum eigenvalue of X.
Since F has a bounded discontinuity at ξ=0, we have that 

    -∞ < F_*(ξ,X) ≤ F^*(ξ,X) < ∞,

for any (ξ, X)∈^N×^N.
Note that F is uniformly elliptic, in the case p∈ (1,∞), since

    min(1/p',1/p) I≤ A(ξ)≤max(1/p',1/p) I,

and merely (degenerate) elliptic in the case p=1, ∞. Moreover, for ξ≠ 0, F is a linear operator in the variable X.


In the case of (<ref>), (<ref>) and (<ref>) are replaced by the following. We say that an upper semicontinuous function in × (0,∞), u:×(0,∞)→, is a viscosity subsolution of (<ref>) if, for every (x,t,ϕ)∈×(0,∞)× C^2(× (0,∞)) such that u-ϕ attains its maximum at (x,t), then

    ϕ_t(x,t)-_p^G ϕ(x,t) ≤ 0  ϕ(x,t)≠ 0
    ϕ_t(x,t) + F_*(0,^2ϕ(x,t)) ≤ 0 ϕ(x,t) =0,

where F_* is given by (<ref>).

We say that a lower semicontinuous function in ×(0,∞), v:×(0,∞)→, is a viscosity supersolution of (<ref>) if, for every (x,t,ψ)∈× (0,∞)× C^2(× (0,∞)), such that v-ψ attains its minimum at (x,t), then

    ψ_t(x,t) - _p^G ψ(x,t) ≥ 0  ψ(x,t)≠ 0
    ψ_t(x,t) + F^*(0,^2ψ(x,t)) ≥ 0   ψ(x,t)=0,

where F^* is given by (<ref>).

A function u that is both a viscosity subsolution and viscosity supersolution is called a viscosity solution of (<ref>).










Here is the case of the resolvent equation (<ref>). We say that an upper semicontinuous function in , u:→, is a viscosity subsolution of (<ref>) if, for every (x,ϕ)∈× C^2() such that u-ϕ attains its maximum at x, then

    u(x) - ^2 _p^Gϕ(x)≤ 0   ϕ(x) ≠ 0;
    
    u(x) +^2 F_*(0, ^2 ϕ(x)) ≤ 0   ϕ(x)=0,

where F_* is given by (<ref>).













We say that a lower semicontinuous function v:→, is a viscosity supersolution of (<ref>) if, for every (x,ψ)∈× C^2(), such that v-ψ attains its minimum at x, then

    v(x) - ^2 _p^G ψ(x)≥ 0   ψ(x) ≠ 0,
    
    v(x) + ^2 F^*(0, ^2 ψ(x)) ≥ 0   ψ(x) =0,

where F^* is given by (<ref>).
A function u is a viscosity solution of (<ref>) if it is both a viscosity subsolution and a viscosity supersolution.


Observe that, since _p^G is degenerate elliptic, then both Lemmas <ref> and <ref> are valid.





 §.§ Comparison and strong maximum principles



In this section, we provide the ad hoc comparison results that will be applied in the rest of the thesis. 







Let  be a domain in ^N, with non-empty boundary . Let u and v be two bounded viscosity solutions of (<ref>) in ×(0,∞). Assume that u and v are continuous on × (0,∞) and on ×{0}.
Then, if u≤ v on × (0,∞)∪×{0}, it holds that

    u≤ v   × (0,∞).




It is enough to observe that we can apply Theorem <ref> to (<ref>). Indeed, (i), (ii) and (iii) of Theorem <ref> are satisfied, thanks to Remark <ref> and the assumption for u-v on the parabolic boundary. On the other hand,  (a) and (b) of Theorem <ref>, follow from (<ref>) and Remark <ref>, since, in the case of F_* and F^* given by (<ref>) and (<ref>), both F_* and F^* are continuous in their variables.


A standard obvious corollary to comparison principle is the uniqueness of solutions of initial-boundary problems. We state the one of our interest.



Let  be as in Corollary <ref>. Then, the bounded viscosity solution of (<ref>)-(<ref>) is unique.


The comparison principle for equation (<ref>) follows from Theorem <ref>, as follows.



Assume that  is a domain, with non-empty boundary . Let u and v be two bounded viscosity solutions of (<ref>) in . Assume that u and v are continuous up to the boundary .
Then, if u≤ v on , it holds that

    u≤ v   .




We need only to ensure that we can apply Theorem <ref> to the case in which F is given by (<ref>). In virtue of Remark <ref>, since F_* and F^* given by (<ref>) and (<ref>) are continuous, it suffices to verify (c) of Theorem <ref>.  The condition (c) of Theorem <ref> is fulfilled since, away from ξ = 0, F  is differentiable with respect to ξ and then, in particular, F is Lipschitz continuous (with respect to ξ) in the compact set {(ξ,X):ρ≤ |ξ| ≤ R, |X|≤ R}. 




Let  be as in Corollary <ref>. Then, the bounded viscosity solution of (<ref>)-(<ref>) is unique.


We give, as corollary of Theorem <ref>, the following result. We observe that in <cit.> the strong minimum principle for (<ref>) is proved, by means of a weak Harnack inequality.



Let  be connected. Let u be a viscosity subsolution of (<ref>) in × (0,∞). If u attains its maximum at a point (x_0,t_0)∈×(0,∞), then u must be constant in × [0,t_0].



We need to check that F in (<ref>) verifies assumptions (a)-(d) of Theorem <ref>. Conditions (b) and (d) are fulfilled by choosing φ () =, since -_p^G is one-homogeneous.
Given s, ξ≠ 0, condition (a) can be read, in the case of game-theoretic p-laplacian, as

    p s -{( I + (p-2) ξ⊗ξ/|ξ|^2)
    ( I - ξ⊗ξ)
    }
    =
    ps - ( N + p - 2 ) +  (p-1) |ξ|^2 > 0,










which is true for any > _0, where _0 = 
[
 N+p-2 - ps/ (p-1)|ξ|^2 ]_+ and for t ∈, [ t ]_+ = max{ 0 , t }.
Analogously, condition (c) it is satisfied, since here we have that

    1 + F_*(0, I)=
    1 -  N / p' >0,

for any < p'/N.




From Remark <ref> we have that also a strong minimum principle holds for (<ref>).



We shall also need an analogous result for equation (<ref>), as application of Theorem <ref>. The proof is straightforward and follows the same line as that of Corollary <ref>.



Let  be connected and u be a nonnegative (viscosity) solution of (<ref>) in . If u attains a maximum at an interior point, then u must be constant in .



We apply Theorem <ref> with G(u,  u, ^2 u ) = u + ^2 F (  u, ^2 u ), where F is given by (<ref>). Indeed, conditions (a) and (c) of Theorem <ref> follow from the uniform ellipticity of -_p^G u and the condition (b) is due to the one-homogeneity of -_p^G u.




Observe that we can apply Theorem <ref> also to G(u, u,^2 u) = F( u,^2 u) where F is given by (<ref>). This gives the strong maximum principle for subsolutions of -_p^G u=0.





 §.§ Hopf-Oleinik lemma



We conclude this chapter with a version of Hopf-Oleinik boundary point lemma. This lemma will be used in Chapter <ref>. In the recent work <cit.> it has been given a sharp version of it, involving the so-called pseudo-balls, defined as follows. Given a, b , R>0 and :[0,R] → [0,∞), the pseudo-ball, 𝒢_a,b^, with apex 0, direction e_N=(0,…,1)∈^N-1 and shape function , is defined by

    𝒢_a,b^ = {x∈ B_R: a(|x|)|x| ≤ x_N ≤ b}.


In <cit.> it is considered the case of classical solutions of certain second order linear differential equations in a domain satisfying a pseudo-ball interior condition. Here, we give its adaptation only to the case of our interest.

We say that :[0,R] → [0,∞) is a Dini continuous function, if it is continuous, (t) > 0, for t ∈ (0,R], and satisfies

    ∫_0^R(t)/t dt<∞.




Let a, b, R >0 and = 𝒢_a,b^, as in (<ref>).
Assume that  is continuous on [0,R], (t)>0, for t∈ (0,R], it satisfies 

    sup_0<t≤ R((t/2)/(t))<∞,

and there exists η > 0, such that

    (s)/s≤η(t)/t  0 < t ≤ s ≤ R.

Suppose that there exists a Dini continuous function :[0,R)→[0,∞), such that

    lim sup_∋ x→ 0[
    (|x|)/|x|x_N/(|x|)]
    <∞.

Let p∈ (1,∞) and u be a viscosity subsolution of

    -_p^G u = 0,

such that  u(0) > u in .
If it is assumed that u is differentiable at 0, then u(0)≠ 0.


The proof follows that of <cit.>. We report its main steps.


Let K > 0 be the finite number that realizes the supremum in (<ref>). As in <cit.>, given > 1+max{0, log_2 K}, C_0, C_1 > 0, we define the function v:𝒢_a,b^→, by

    v(x)  = - x_N - C_0∫_0^x_N∫_0^(t)/t  dt   d + C_1 ∫_0^|x|∫_0^ (t)/t(t/ξ)^ - 1  dt   d.

We first list the relevant proprieties of v, which follow by the assumptions on both  and . The function v is of class C^0()∩ C^2(), it satisfies v(0)=0 and v(0) = -e_N.
By combining (<ref>), assumptions on  and elementary integral inequalities, as shown in <cit.>, it holds that

    v(x) ≥( C_1/2η -a - a C_0 ∫_0^b(t)/t dt ) |x|(|x|)   𝒢_a,b^∖{x_N = b }.

For a fixed C_1 > 2 a η, it is sufficient to choose b_* > 0 such that

    ∫_0^b_*(t)/t dt ≤C_1 - 2 a η/2 a η C_0
 
to obtain that v ≥ 0 on 𝒢_a,b_*^∖{x_n = b_*}. Moreover, since u and v are continuous functions and the fact that u(0) > u in 𝒢_a,b^, on = { x∈𝒢_a,b^: x_N = b_* }, there exists > 0 such that

    < min_|u-u(x_0)|/max_|v|.

In particular, this last condition implies that v ≥ u - u(x_0) on .
Hence, there exist b_* >0 and > 0 such that 

    v ≥ u - u(0)  𝒢_a,b_*^.
 
Now, in order to apply comparison, we only need to show, for some C_0, that v is a supersolution of (<ref>).











By differentiating twice, we have that

    ^2 v(x) = -C_0 (x_N)/x_N e_N⊗ e_N  +
     C_1 ∫_0^|x|(t) t^ - 2  dt[I/|x|^ - x⊗ x/|x|^ + 2]+C_1 (|x|)/|x|x ⊗ x/|x|^2.





Since v(0)= -e_N, by possibly taking R sufficiently small, we can assume that v(x) ≠ 0, for any x∈. After setting ξ(x) =  v (x)/| v(x)|, we can calculate the following:

    p{A(ξ)^2 v}(x)=-C_0 (x_N)/x_N[1+(p-2)ξ_N(x)^2]+
    
    C_1/|x|^∫_0^|x|(t)t^-2 dt{N + (p-2) - - (p-2)  x,ξ(x)^2/|x|^2}+
    
    
    C_1(|x|)/|x|[1+(p-2) x,ξ(x) ^2/|x|^2],

where A(ξ), defined in (<ref>), is the matrix of coefficients of _p^G.
Then, by using also that |ξ (x)| = 1, we infer that

    p_p^G v(x)≤
    (x_N)/x_N[-C_0 c_1,p +C_1/|x|^∫_0^|x|(t) t^-2 dt x_N/(x_N)[N+p -2 - c_1,p]_+ +. 
    . C_1(|x|)/|x|x_N/(x_N)c_2,p],

where c_1,p=min{1, p-1}, c_ 2,p = max{ 1, p-1 } and [t]_+ = max{0 , t }. 

The previous inequality gives that

    _p^G v (x)≤
    
    (1/p)(x_N)/x_N{ -C_0 c_1,p+C_1 (M[N+p-2- c_1,p]_+ + c_2,p) },

where

    Λ=sup_x∈((|x|)/|x|x_N/(x_N)),

and

    M=sup_x∈(|x|^1-/(|x|)∫_0^|x|(t)t^ - 2 dt).

Observe that  and M are finite as shown in the proof of <cit.>, from the assumptions on  and .
Thus, by choosing C_0 such that 

    C_0 > C_1 M /c_1,p[N+p-2 - c_1,p]_+ + c_2,p/c_1,p,

then _p^G v (x) ≤ 0. Hence, by the arbitrariness of x, we have that -_p^G v ≥ 0 in 𝒢_a,b_*^.




Now, from

    v ≤ u - u(0)   𝒢_a,b_*^

and v(0) = -e_N, we can readily obtain the conclusion by a standard argument.





























In <cit.>, it has been observed that the conditions in Lemma <ref> on  are sharp (see <cit.> for details).





The class of domains for which Lemma <ref> is valid is quite large and it is given implicitly by the assumptions that must be satisfied by .
In particular, if  is a domain of class C^1,, for some ∈ (0,1), by choosing (t) = (t) = t^, the conclusion of Lemma <ref> holds true. 
In <cit.>, it has been observed that, if  is a domain of class C^1,, where  is Dini continuous and quasi-increasing, then the conclusion of Lemma <ref> holds true. We say that  is quasi-increasing if there exists η̃ > 0 such that (s) ≥η̃ (t), for any 0 < s ≤ t.



In order to apply Lemma <ref> to our case, we need to show that the solution of (<ref>)-(<ref>) is a (viscosity) subsolution of the equation (<ref>) on every slice ×{t}, for t > 0.




Let u be the viscosity solution of (<ref>) satisfying (<ref>)-(<ref>). 
Then, for every t > 0, u(·, t) : → is a viscosity subsolution of (<ref>) in . 



Given τ > 0, set

    v(x,t) = u(x,t + τ)  (x,t) ∈× (0,∞).

Up to translate the test functions in (<ref>) and (<ref>), we verify that v is a viscosity solution of (<ref>). After that, the boundary condition (<ref>) is obviously satisfied by v, and we have that v(x, 0 ) = u(x,τ) > 0, by applying the strong minimum principle (Remark <ref>). Corollary <ref> gives that v ≥ u on ×(0,∞), which yields the following inequality

    u (x,t+τ) ≥ u(x,t),  x∈,  t,τ >0.

Now, for a fixed t>0, let (x,φ) ∈× C^2() such that u(·,t) - φ attains its maximum at x. We show that 

    F_*(φ (x), ^2 φ(x) ) ≤ 0,

where F is that in (<ref>).
Define the function ϕ∈ C^2(×(0,∞)) by ϕ(y,s) = φ(y), for (y,s)∈× (0,∞). From (<ref>) and the assumption on φ, it follows that u(x,t) - ϕ(x,t) ≥ u(y,s) - ϕ(y,s), for any y∈ and 0 < s ≤ t. By proceeding as in <cit.>, this inequality is sufficient to infer that

    ϕ_t(x,t) + F_*(ϕ(x,t),^2 ϕ(x,t)) ≤ 0,
 
since F is a quasi-linear (degenerate) elliptic operator. Therefore, (<ref>) follows and this concludes the proof.


We say that  satisfies an interior -pseudo-ball condition in x_0∈, if there exist a, b, R >0 and a modulus , such that, up to translate and rotate , 𝒢_a,b^⊂ and 𝒢_a,b^∩ ={x_0}.


[Hopf-Oleinik lemma for (<ref>)]

Let  be a domain satisfying the interior -pseudo-ball condition, with  that satisfies assumptions of Lemma <ref>. 
Let p∈ (1,∞) and u be the viscosity solution of (<ref>)-(<ref>). Assume that there exist x_0∈ and t > 0, such that  

    u(x_0,t) > u  𝒢_a,b^×{t}.

If it is assumed that u is differentiable at x_0, then u(x_0,t)≠ 0.



We apply first Lemma <ref> and then Lemma <ref> to y ↦ u(y,t).




[Hopf-Oleinik lemma for (<ref>)]

Let  be a domain satisfying the interior -pseudo-ball condition, with  that satisfies assumptions of Lemma <ref>. 
Let p∈ (1,∞) and u^ be the viscosity solution of (<ref>)-(<ref>). Assume that there exists x_0∈, such that  

    u^(x_0) > u  𝒢_a,b^.

If it is assumed that u^ is differentiable at x_0, then u^(x_0)≠ 0.



Observe that, by setting w ≡ 0 on , then w is a solution of (<ref>) and w < u^ on . Hence, by applying Corollary <ref> we have that u^≥ 0 on . This implies that u^ is a viscosity subsolution of (<ref>). We then conclude by applying Lemma <ref>.












CHAPTER: ASYMPTOTICS FOR EXPLICIT SOLUTIONS



In this chapter, we consider equations (<ref>) and (<ref>) and we deduce asymptotic formulas for global solutions and for solutions of certain boundary-value problems in symmetric domains, such as the half-space, the ball and the exterior of a ball. 

The obtained solutions will later be used as barriers to extend the relevant asymptotic formulas to more general domains.
In the cases examined in this chapter, the (viscosity) solutions can be explicitly computed by taking advantage of the fact that they are smooth and that the relevant equations become linear. The corresponding boundary-value problems become considerably simpler, since they concern functions that depend on only one space variable.
Most of the explicit representations are based on Bessel functions, whose relevant properties are recalled in Section <ref>. We present the corresponding theorems in Sections <ref>, <ref> and <ref>. Section <ref> is then devoted to the asymptotic analysis of the obtained solutions as the relevant parameters tend to zero.

























§ FORMULAS FOR BESSEL FUNCTIONS


For an overview of this subject, we refer to <cit.>. Here, we briefly collect the properties of our interest. 
Given ∈, the Bessel's equation of order , is the following ordinary differential equation:

    ^2 y” +  y' +(^2-^2) y =0    > 0.

 Every solution of (<ref>) can be written as

    A   J_() + B  Y_(),

where J_ is called a Bessel function of first kind, Y_, is called a Bessel function of second kind, and A, B are constants. We know that J_ is finite at =0, while Y_ is singular at =0.









The following result can be found in <cit.>.


Let p ∈ (1,∞). The eigenfunctions of the problem

    v”() + N-1/p-1v'()/ + p/p-1_n v() = 0   (0,1),
       v'(0)= v(1)  = 0,
  
are given by

    v_n() =  ^-N-p/2(p-1) J_N-p/2(p-1)(_n),

where, _n is the n-th zero of J_N - p/2(p - 1) and _n = _n^2/p', n= 1, 2, ⋯.
Moreover, after normalization, the set {v_n: n∈} form a complete orthonormal system in the weighted space L^2((0,1); ^ N - 1 / p - 1 d).




Observe that Lemma <ref> can be extended to the case p=∞. In this case, for n = 1,2,⋯, the eigenfunctions are given by

    v_n () = √(2)cos (_n ),   _n = (2n - 1) π/2.



We will also use the modified Bessel functions, that can be defined by the formulas:

    I_() = i^- J_(i ),   ∈,
       K_()= π/2 sin(π)(I_()-I_-()),   ∈,

where here by J_ we mean the analytic extension to the complex plane. (Notice that the above definitions hold when  is not an integer; at integer points I_ and K_ are obtained as limits in the parameter .)
We say that I_ is a modified Bessel function of first kind and K_ is a modified Bessel function of second kind. They are two linearly independent solutions of the modified Bessel's equation of order :

    ^2 y” +  y' -(^2+^2) y =0.


We will use the following integral representations of I_ and K_ (see <cit.>), 

    I_() = (/2)^/√(π)(+1/2)∫_0^π e^cos(sin)^2 d,   ()> - 1/2,

and

    K_()= √(π)(/2)^/(+1/2)∫_0^∞ e^-cosh(sinh)^2 d, 
      () >-1/2,  |()| <π/2.


The next lemma will be used to obtain solutions related to the game-theoretic p-laplacian in radially symmetric domains and it can be seen as corollary of <cit.>.



Let p ∈ (1,∞) and > 0. The functions

    ^p - N/2(p - 1) I_N - p/2(p - 1)(√(p/p-1))  ^p - N/2(p - 1) K_N - p/2(p - 1)(√(p/p-1)),

are linearly independent solutions of

    y”() + N-1/p-1y'()/ - p/p-1^2 y() = 0   >0.




The proof follows at once, by direct inspection.




















We conclude this section by deducing asymptotic formulas for the integrals involved in (<ref>) and (<ref>). The next lemma is essentially contained in <cit.>.



For > -1 and >0, let

    g() = ∫_0^π e^- (1- cos) (sin)^ d,


    f()=∫_0^∞ e^- (cosh-1) (sinh)^ d.
 
Then, we have that

    g() = 2^ -1/2(+1/2)  ^-+1/2{1+ O(1/)}


    f()=
    2^-1/2(+1/2)^-+1/2{ 1+O(1/)}

as →∞ and

    f()=^- (){ 1+o(1)}     >0, 
    
    
    log(1/)+O(1)      =0, 
    
    
    √(π)/2 sin(π/2) (+1/2)/(/2+1)+o(1)       -1<<0,

as → 0^+.



To start with, we establish formula (<ref>). By the change of variable τ= (1-cos) we get:

    g()=2^-1/2^-+1/2∫_0^∞ e^-τ(τ-τ^2/2)^-1/2dτ.

This formula gives (<ref>), once we observe that the integral ∫_0^∞ e^-ττ^+1/2 converges.
To deduce (<ref>), we use the change of variable τ=  (cosh-1), obtaining

    f()=1/∫_0^∞ e^-τ(2τ/+τ^2/^2)^-1/2dτ.

When →∞, our claim follows, as above, by writing

    f()=2^-1/2^-+1/2∫_0^∞ e^-τ(τ+τ^2/2)^-1/2dτ.

When → 0^+ and >0, our claim follows by writing

    f()=^-∫_0^∞ e^-τ(τ^2+2 τ)^-1/2dτ,

since the integral ∫_0^∞ e^-ττ ^ -1  dτ converges, from > 0.
For -1<≤ 0, we use <cit.>, to infer that

    f()=1/√(π) (+1/2)(/2)^-/2 e^  K_/2().

Then, <cit.> give our claims for =0 and -1<<0, respectively.








§ THE GLOBAL SOLUTIONS



In this section, we present the global solutions of the parabolic equation (<ref>) and of the resolvent equation (<ref>). The former can be found in <cit.> . We recall that p' = p/(p-1) and we mean p'=1 for p=∞.



Let p ∈ (1,∞] and Φ be the function defined as

    Φ(x,t) =
    
     t^-N+p-2/2(p-1)e^-p'|x|^2/4 t  p∈(1,∞),
     
    
    
    t^-1/2 e^-|x|^2/4t  p=∞,

for x∈^N and t > 0.
Then, Φ is a viscosity solution of (<ref>) in ^N × (0,∞) and is bounded on (^N∖ B_) × (0,∞), for > 0.


Observe that Φ∈ C^∞( ^N ×(0,∞) ) and that Φ(x,t) = ϕ(|x|,t), where ϕ=ϕ(r,t) is clearly defined by (<ref>). Equation (<ref>) for a radially symmetric function u reads as

    u_t=p-1/p u_rr + N-1/pu_r/r.
 
By simple computations, we verify that this equation is satisfied (pointwise) by ϕ(r,t) for r 0 and t>0.
Thus, Φ is a classical solution in (^N ∖{0}) × (0,∞), where its spatial gradient does not vanish. Then, we apply Lemma <ref> to conclude.
The stated boundness of Φ easily follows by its definition.




Let p ∈ (1,∞] and Φ_ be the function defined, for x∈^N, by

    Φ_(x) = 
    ∫_0^∞e^-√(p')|x|/cosh(sinh)^ N - p / p - 1  d     p∈(1,∞), 
    
    
    
    e^-|x|/     p=∞.

Then Φ_ is a viscosity solution of (<ref>) in ^N∖{ 0}. Moreover, in the case p∈(N,∞], Φ_ is bounded in ^N while, in the case p∈ (1,N], it is bounded on the complement of any ball centered at 0.



The case p=∞ is immediate. In the case p∈(1,∞), from (<ref>), we have that Φ_ = ϕ(|x|), where ϕ(r) is a solution of (<ref>), for r > 0. A direct check shows that (<ref>) is simply the equation (<ref>) for radial functions. Hence, Φ_ is a solution of (<ref>), outside the origin. The stated boundedness follows from the properties of the function K_N-p/2(p-1).








§ ELLIPTIC SOLUTIONS IN SYMMETRIC DOMAINS

























Set p∈ (1,∞].
Then, for x∈ B_R, the following function,

    u^(x)=∫_0^πe^√(p')|x|/cos(sin) ^ N - p / p - 1  d/∫_0^πe^√(p')R/cos(sin) ^ N - p / p - 1  d     p∈(1,∞),
    
    
    cosh(|x|/)/cosh(R/)     p=∞,

is the (viscosity) solution of (<ref>)-(<ref>).



We only consider the case p ∈ (1,∞), while the extremal case p=∞ is similar and simpler. We have that u^(x) = ϕ(|x|), where ϕ(r) = r^p-N/2(p-1)I_N-p/2(p-1)(r). For Lemma <ref>, ϕ(r) is a solution of (<ref>), which is the equation (<ref>) for radial functions. Hence, u^ is a classical solution of (<ref>), away from the origin. Since u^ is of class C^2 in B_R, then Lemma <ref> informs  us that it is also a viscosity solution in the whole ball. Finally, it is clear that u^ = 1 on the boundary, then it is the solution of (<ref>)-(<ref>), by uniqueness (Corollary <ref>).


























Set p ∈ (1, ∞].
Then, the following function, defined for x∈^N ∖ B_R, by

    u^(x)=
    ∫_0^∞e^-√(p')|x|/cosh(sinh)^ N - p / p - 1  d/∫_0^∞e^-√(p')R/cosh(sinh)^ N - p / p - 1  d     p∈(1,∞),  
    
    
    
    e^-|x|-R/     p=∞,

is the bounded (viscosity) solution of (<ref>)-(<ref>).



We just need to observe that u^ is just Φ_ of Proposition <ref>, once it is normalized to the value 1 on the boundary of B_R. Hence, u^ is a bounded solution of (<ref>)-(<ref>). By uniqueness (Corollary <ref>), we conclude.


















































































§ PARABOLIC SOLUTIONS IN SYMMETRIC DOMAINS











Before going on, we point out that, from now on, with  we indicate the boundary of the relevant set, which it will be evident by the context.
First, we focus our attention on the case of the half-space of ^N. We will use the complementary error function defined by 

    ()=2/√(π) ∫_^∞ e^-τ^2 dτ,   ∈.




Let p∈ (1,∞] and H be the half-space in which x_1>0. The function Ψ, defined by 

    Ψ(x,t)=√(p'/4π)∫_x_1/√(t)^∞ e^-1/4 p' ^2 d=
    (√(p') x_1/2√(t))   (x,t)∈H× (0,∞).

is the bounded solution of (<ref>)-(<ref>).



In virtue of Corollary <ref>, the bounded solution of (<ref>)-(<ref>) is unique. The function Ψ is smooth with no critical points. To conclude, it is enough to note that, after an inspection, it satisfies pointwise (<ref>) in H × (0,∞) and both (<ref>) and (<ref>).





Now, we establish a series representation of parabolic solutions in the ball.



Let p ∈ (1,∞].
Then, the solution of (<ref>)-(<ref>) has the following representation:


  * if p ∈(1, ∞), we have that, for x∈ B_R and t > 0,

    u(x,t) = 2 (  R / |x| )^ N- p / 2(p - 1) ∑_ n = 1 ^∞ J_ N - p / 2(p - 1) ( _n / R |x| ) /_n   J_ N + p - 2 / 2(p - 1) ( _n ) ( 1 - e^-_n^2/p'  R^2 t).

where _n is the n-th positive zeros of J_N - p/2(p - 1). 


  * If p=∞, we get, for x∈ B_R and t > 0:

    u(x,t)=
    4/π∑_n=1^∞ (-1)^n-1cos( (2n-1)π |x|/2R) / 2n-1 ( 1 - e^-(2n-1)^2 π^2/4R^2 t).





Preliminarily, we observe that, since (<ref>) is invariant if we add a constant to u, 1 - u is also a solution of (<ref>). Moreover, 1 - u vanishes on × (0,∞) and is equal to 1 on B_R×{0}. We want to find the series representation of 1 - u.
We use the same scheme, in all cases p ∈ (1,∞], clarifying that, in the extremal case p = ∞, we use Remark <ref> instead of Lemma <ref>.
Now, set = N-p/2(p-1).
By applying Lemma <ref>, we infer that {^- J_( _n/R):n=1,2,⋯} is a complete system in  L^2((0,R); ^2 +1  d). This fact implies that 

    ∑_n=1^∞ c_n {^- J_(_n/R)} = 1   ∈ (0,R),

for

    c_n = 
    ∫_0^R^ + 1 J_(_n/R)   d/∫_0^R J_( _n/R)^2  d
    =
    ∫_0^R^ + 1 J_(_n/R)   d/ R^2 J_ + 1(_n)^2 /2,

where, in the last equality, we have used <cit.>. 
Now, since, by formula <cit.> we know that

    ∫_ 0 ^ 1 ^ + 1  J_( _n )   d =  J_ + 1 (_n) /_n ,

we obtain

    c_n = 2 R^/_n J_ +1 (_n).

We see that, for any n=1,2,⋯, by reason of (<ref>), the function

    u_n (x,t) = |x|^- J_(_n/R |x|) e^-_n^2 / p' R^2  t,

 is a solution of (<ref>) in B_R× (0,∞) and it vanishes on × (0,∞). 
 Thus, by uniqueness (Corollary <ref>, we have that

    1 - u(x,t) = ∑_n=1^∞2 R^/_n J_ + 1(_n) |x|^- J_(_n/R |x|) e^-_n^2 / p' R^2  t,

which, by using (<ref>), gives (<ref>). 
































Finally, we observe that, thanks to Remark <ref>, in the case p = ∞, (<ref>) is just (<ref>).


Last, we report the following connection between elliptic and parabolic radial (viscosity) solutions, as stated in <cit.>.



Let p ∈ (1,∞]. Let u(x,t) be given by Lemma <ref>. Then, it holds

    u^(x) = ^-2∫_0^∞ u(x,τ) e^-τ/^2 dτ  x∈B_R,

where u^ is the solution of (<ref>)-(<ref>), in B_R.



We prove that both sides of (<ref>) have the same eigenfunction expansion. Here, we treat the case p∈ (1,∞). In the extremal case p=∞ we have only to utilize the expressions of eigenfunctions in Remark <ref>.
Let _n be given by Lemma <ref>, then, for any >0, it holds that

    ^-2∫_0^∞(1 - e^_n^2 τ/p'R^2) e^-τ/^2 dτ
     = 
    ^2_n^2/(p'R^2)/^2_n^2/(p' R^2) +1 
    =
     ^2 _n^2/^2 _n^2+ p' R^2.

Hence, from (<ref>), we obtain that the right-hand side of (<ref>) equals

    2 (R/|x|)^N-p/2(p-1) R^- 2∑_ n=1 ^∞_n / J_ N + p - 2 / 2(p - 1) (_n) {p'/^2 + _n ^2/R^2}^-1  J_N-p/2(p-1)(_n/R|x|),

for any x∈ B_R and > 0. 
Now, we observe that the sum of the last series is the function u^. Indeed, by comparing (<ref>) and (<ref>), we have that u^ is given by 

    u^(x) = 
    (R/|x|)^ N - p / 2(p - 1)   I_ N - p / 2(p - 1) (√(p')/|x|) / I_N-p/2(p-1)(√(p')/R)

and then the relevant coefficients can be calculated by










 applying <cit.>, that is

    ∫_0^R I_N-p/2(p-1)(√(p')/)  J_N-p/2(p-1)(_n/R) d =
    _n I_N-p/2(p-1)(√(p') R/) J_N+p-2/2(p-1)(_n){p'/^2 + _n ^2/R^2}^-1,

which gives (<ref>).




















































































§ ASYMPTOTICS



In this section, we collect asymptotic formulas for the functions presented in Sections <ref> and <ref>.



 §.§ The elliptic case



 

Let p∈ (1,∞]. Assume that u^ be the (viscosity) solution of (<ref>)-(<ref>) in B_R.
Then, it holds that

    log u^+√(p') d_=
     
    O( log)      1<p<∞,
    
    
    O()       p=∞,

uniformly on B_R as → 0^+.



First, we observe that d_ ( x ) = R - |x|.
The case p=∞ follows at once, since (<ref>) gives:

    log{u^(x)}+d_(x)=log[1+e^-2|x|//1+e^-2R/].

If 1<p<∞, by (<ref>) we have that

    log{u^(x)}+√(p') d_(x)=
     log[∫_0 ^πe^-√(p')(1-cos) |x|/(sin)^N-p/p-1 d/∫_0^πe^-√(p')(1-cos) R/(sin)^N-p/p-1 d]

and the right-hand side is decreasing in |x|, so that 

    0
    ≤ log{u^(x)}+√(p') d_(x)≤ log[
    ∫_0^π(sin)^N-p/p-1 d/∫_0^πe^-√(p') (1-cos) R/(sin)^N-p/p-1 d].

This formula gives (<ref>), since we have that 

    ∫_0^πe^-√(p') (1-cos)R/(sin)^N-p/p-1 d=2^N - 2p  +1/2(p-1)(N - 1/2p-2)(R√(p')/)^-N - 1/2p - 2 [1+O()]

as → 0^+, by using (<ref>), with = √(p') R/.


The next theorem is contained in <cit.>.


Let p ∈ (1,∞]. Assume that u^ be the (viscosity) solution of (<ref>)-(<ref>) in ^N∖B_R.
Then, it holds that

    log u^+√(p') d_=O()    → 0^+,

uniformly on every compact subset of ^N∖ B_R.



Notice that d_(x)=|x|-R. If p=∞, (<ref>) holds exactly as

    log{u^(x)}+d_(x)≡ 0.

If 1<p<∞, we write that

    log{u^(x)}+√(p') d_(x)=
    log[
    ∫_0^∞e^-√(p')|x|/(cosh-1)(sinh)^ N - p / p - 1  d/∫_0^∞e^-√(p')R/(cosh-1)(sinh)^ N - p/ p - 1  d]

and hence, by monotonicity,  we have that

    log{∫_0^∞e^-√(p')R'/(cosh-1)(sinh)^N-p/p-1 d/∫_0^∞e^-√(p')R/(cosh-1)(sinh)^N-p/p-1 d}≤log{u^(x)}+√(p') d_(x)≤ 0,

for every x such that R≤|x|≤ R', with R'>R.
Our claim then follows by an inspection on the left-hand side, after applying (<ref>), with =√(p') R'/ and =√(p') R/.




 §.§ The parabolic case



We state the following theorem, contained essentially in <cit.>.



Let p ∈ (1, ∞] and let H be the half-space in which x_1 > 0. Let Ψ be given by 

    Ψ(x,t)=√(p'/4π)∫_x_1/√(t)^∞ e^-1/4 p' ^2 d,

for x∈ H and t >0.
Then, it holds that 

    4tlog{Ψ(x,t)} = -p' d_(x)^2 + O(t log t),

uniformly for x in every strip { x∈^N: 0≤ x_1≤} with >0.



Observe that d_(x) = x_1. By employing a change of variables in the expression of Ψ, we get that

    Ψ(x,t)=e^-p' x_1^2/4t √(p'/4π)∫_0^∞ e^-1/2 p' x_1/√(t) -1/4 p' ^2 d,

and hence

    4t logΨ(x,t)+p' x_1^2= 4t log( √(p'/4π)∫_0^∞ e^-1/2 p' x_1/√(t) -1/4 p' ^2 d).

Thus, for 0≤ x_1≤, we have that

    4t log( √(p'/4π)∫_0^∞ e^-p' /2√(t)-p' ^2/4 d)
    ≤
     4t logΨ(x,t)+p' x_1^2≤ 0.

Integrating by parts on the two functions

    e^-p'/2√(t)  e^-p' ^2/4

and a change of variable give that

    ∫_0^∞ e^-p' /2√(t)-p' ^2/4 d 
    =
    2/ p'√(t)[ 1 + O(t) ]











as t → 0^+  (see also <cit.>). Then (<ref>) gives the desired uniform convergence.



In the case of the ball, the series representation of u(x,t), established in Lemma <ref>,
is not convenient to obtain an asymptotic formula for t→ 0. We then proceed differently. To start with, we state the next lemma. 



Let p ∈ (1,∞]. Assume that u is the solution of (<ref>)-(<ref>).
If p ∈ (1,∞), then, for every x ∈B_R, t > 0 and > 0, it holds that

    4t log u(x,t) 
    ≤4/^2 -
    
    4/√(p') d_(x)
    +
    4 t log[
    ∫_0^π (sin)^N-p/p-1  d/∫_0^π e^-√(p')/ t (1-cos) d_(x) (sin)^N-p/p-1 d].















If p=∞, for every x∈B_R, t > 0 and > 0, we have that

    4tlog u(x,t) ≤4/^2 - 4/d_(x) + 4t log[ 2/1 + e^-2 d_(x)/ t].




Preliminarly, observe that if x∈, (<ref>) and (<ref>) are obviously satisfied. 
For a fixed x∈ B_R, we argue as follows. By  Lemma <ref>, for every >0 the function
u^ defined in (<ref>) is the solution of (<ref>)-(<ref>) in B_R. Moreover, from (<ref>) in the proof of Lemma <ref>, we have that t↦ u(x,t) is increasing.
Thus, it holds that

    ^2 u(x,t) e^-t/^2≤∫_t^∞ u(x,τ) e^-τ/^2 dτ≤∫_0^∞u(x,τ)e^-τ/^2 dτ = ^2 u^(x),

and hence

    u(x,t)≤ u^(x) e^t/^2;

the last inequality holds for any t, >0. Next, we choose
=  t and obtain that

    u(x,t)≤ u^ t(x) e^1/(^2 t)  t>0.

Therefore, 

    4t log u(x,t)≤ 4t log u^ t(x)+4/^2.

Now, let B^x=B_d_(x)(x), be the ball centered at x, with radius d_(x). Let u_B^x^ be the solution of (<ref>)-(<ref>) in B^x. By the comparison principle (Corollary <ref>), we have that, for any t, > 0 and x∈ B_R,

    u^ t≤ u_B^x^ t  B^x

and, in particular,  

    u^ t(x) ≤ u_B^x^ t(x).

By using this fact and the previous inequality, we obtain that, for any x∈ B_R, t>0 and >0,



    4t log u(x,t)≤ 4t log u_B^x^ t(x)+4/^2,

which, together with  (<ref>), gives (<ref>) and (<ref>). Indeed, we observe that, by translating, u^ t_B^x(x) is simply the solution of (<ref>)-(<ref>) in B_d_(x), with = t, evaluated at the origin. 


We are ready to obtain the uniform asymptotics for u(x,t) in the ball. The pointwise formula is given in <cit.>.



Set p ∈ (1,∞] and let u(x,t) be the viscosity solution of (<ref>)-(<ref>). 
Then, it holds that

    4tlog u(x,t) + p' d_(x)^2 = O ( t log t),

as t → 0^+, uniformly on B_R.



Given x∈ B_R there exists y∈ such that |x-y|=d_(x) (y is unique unless x=0). Let H be the half-space containing B_R and such that H∩={ y}; notice that d_(x)=d_ H(x).
Let Ψ^y be the solution of (<ref>)-(<ref>) in H×(0,∞); since B_R is contained in H, Ψ^y obviously satisfies (<ref>) and (<ref>) for B_R and, also, Ψ^y≤ 1 on ×(0,∞).
By the comparison principle (Corollary <ref>), we get that u≥Ψ^y and hence

    4tlog u(x,t)≥ 4tlogΨ^y(x,t)   (x,t)∈B_R×(0,∞).

Thus,  Theorem <ref> with = 2R, implies that

    4tlog u(x,t)≥ -p' d_ H(x)^2 + O(t log t),

uniformly on B_R, as t → 0^+, which is (<ref>) by one side, since d_ H(x) = d_(x).
On the other hand, by applying Lemma <ref> with  = ^*>0 such that 

    -4 √(p') d_(x)/^*+4/(^*)^2=-p' d_(x)^2,   ^*=2/√(p')  d_(x),

we obtain, if p∈(1,∞),

    4 tlog u(x,t)≤ -p' d_(x)^2 + 4  t log[
    ∫_0^π (sin)^N-p/p-1  d/∫_0^π e^-p'/2 t (1-cos) d_(x)^2 (sin)^N-p/p-1 d],

and the obvious corresponding inequality, in the case p=∞.
Hence, by applying (<ref>) with = p' d_(x)^2 / (2  t), we conclude that

    4 tlog u(x,t)≤ -p' d_(x)^2 + O ( tlog t),

uniformly on B_R, as t → 0^+. Putting together (<ref>) and (<ref>), we conclude the proof.







 


CHAPTER: VARADHAN-TYPE FORMULAS






In his paper <cit.>, S.R.S. Varadhan considered the following problems for the heat and resolvent equations: 

    u_t - 1/2[A  ^2 u] = 0     ×(0,∞),
       u=1     × (0,∞),
       u=0      ×{0},





and

    ^-2 u^ - 1/2[ A  ^2 u^]=0     ,
       u^=1     ,





where A=A(x), for x∈, is a symmetric and positive definite N× N matrix.
When A is uniformly elliptic and uniformly Hölder continuous, Varadhan proved the following asymptotic formulas: 

    -lim_t→ 0^+2tlog u(x,t) = d_^A(x)^2

and

    -lim_→ 0^+log u^(x) = √(2) d_^A(x),

where d_^A is the distance from the boundary of , induced by a Riemannian metric related to the matrix A. In particular, when A is the identity matrix, d_^A=d_ coincides with the usual euclidean distance, defined by

    d_(x) = inf{|x-y|:y∈}.

In this chapter, we establish asymptotic Varadhan-type formulas for the solutions of problems (<ref>)-(<ref>) and (<ref>)-(<ref>), in quite general domains. See Theorems <ref> and <ref>.
In particular, we prove the pointwise formulas (<ref>) and (<ref>), that is

    -lim_t→ 0^+ 4 tlog u(x,t) = p' d_(x)^2

and 

    -lim_→ 0^+log u^(x) = √(p')  d_(x).

These formulas hold for a (not necessarily bounded) domain  which merely satisfies the topological assumption = (^N ∖). 
We also compute the uniform rate of convergence in (<ref>) and (<ref>). In fact, we show that

    4 tlog u(x,t) + p' d_(x)^2=O(tlogψ(t))   t→ 0^+,

 and, for → 0, that
 
    log u^(x) + √(p') d_(x) = 
      
      O()   p=∞,
     
      O( log)   p ∈ (N, ∞),

and
  
    log u^(x) +√(p')  d_(x)=
       
      O( log |logψ()|)  p=N,
      
    
      O(logψ())   p∈(1,N),

uniformly on every compact subset of . The function ψ depends on appropriate regularity assumptions on . See Theorems  <ref> and <ref>.
The results presented in this chapter are based on the construction of suitable barriers, which essentially employ the radial solutions, deduced in Chapter <ref>. In Sections <ref> and <ref>, we present such barriers.  We stress the fact that no regularity assumption on  is needed. 
Sections <ref> and <ref> are then dedicated to deduce the already mentioned pointwise and uniform asymptotic formulas.




§ BARRIERS IN THE PARABOLIC CASE






































The next two lemmas give global barriers from below and above for the solution of (<ref>)-(<ref>). 



Set 1<p≤∞. Let ⊂^N be a domain and let z∈^N∖. Assume that u(x,t) is the bounded (viscosity) solution of (<ref>)-(<ref>).
Then, for every  x∈ and t>0, it holds that

    4 tlog{u(x,t)} + p' |x - z|^2
     ≥
     4 t  log E^-(d_(z),t).
 
Here, for > 0 and t > 0, we mean that

    E^-(,t) =
    A_N,p t^- N + p -2 / 2(p - 1) ^ N + p - 2 / p - 1 ,

where

    A_N,p = {pe/2(N + p -2)}^N + p -2/2(p - 1).




The function U^z, defined, for x∈ and t>0, by

    U^z(x,t) = { A_N,p d_(z)^N+ p -2/p-1}Φ(x-z,t)   (x,t) ∈× (0,∞),

 is a solution of (<ref>) in ×(0,∞), since Φ is a global solution (Proposition <ref>). Moreover, U^z(x,0^+)=0, for any x∈ and, in virtue of the fact that

    max{Φ(x-z,t): x∈,  t>0 } = (A_N,p d_(z)^N+p-2/p-1)^-1,

we also have that U^z ≤ 1 on ×(0,∞). 
Hence, we just apply the comparison principle (Corollary <ref>), to conclude that U^z ≤ u, on × (0,∞), which implies  (<ref>), by recalling the definition of Φ.




Set 1 < p ≤∞. Let ⊂^N be a domain. Let u be the bounded (viscosity) solution of (<ref>)-(<ref>).
Then, for every x∈ and t > 0, it holds that

    4tlog{u(x,t)} +p' d_(x)^2
    ≤
    4 t log E^+(d_(x), t),

where E^+(, t) is given by

    E^+(, t)=
    ∫_0^π (sin)^N-p/p-1  d/∫_0^π e^-p'/2 t (1-cos) ^2 (sin)^N-p/p-1 d     p∈(1,∞),
    
    
    2/1 + e^-p'/t^2     p=∞,

for ≥ 0 and t>0.
In particular, it holds that

    t log E^+(d_(x), t)=
    
    O(t log t)   p ∈ (1,∞),
    
    O( t)  p=∞,

as t→ 0^+, uniformly on every subset of  in which d_ is bounded.



Let u_B be the solution of  (<ref>)-(<ref>) in the unit ball B. We prove that 
u(x,t)≤ u_B(0,t/d_(x)^2),
for (x,t)∈×(0,∞), where we mean that  u_B(0,t/d_(x)^2)=1 when x∈. 
Indeed, for x∈, the inequality is satisfied as an equality. Let x∈ and let v^x=v^x(y,t) be the solution of (<ref>)-(<ref>) in B^x×(0,∞), where B^x is the ball centered at x with radius d_(x). Corollaries <ref> and <ref> give that

    u(y,t)≤ v^x(y,t)  (y,t)∈B^x×(0,∞),
 
and hence, in particular, u(x,t)≤ v^x(x,t) for every t>0. Since x is arbitrary in , we infer that

    u(x,t)≤ v^x(x,t)  (x,t)∈×(0,∞).

Now, for fixed x∈, consider the function defined by 

    w(y,t)=v^x(x+d_(x) y,d_(x)^2 t)  (y,t)∈B×(0,∞).

By the translation and scaling invariance of (<ref>), we have that w satisfies the problem (<ref>)-(<ref>) in B, and hence equals u_B on B×(0,∞), by uniqueness.
Therefore, evaluating u_B at (0, t/d_(x)^2) gives that

    u_B(0,t/d_(x)^2)=w(0,t/d_(x)^2)=v^x(x,t)≥ u(x,t),

by (<ref>). 
We conclude and obtain (<ref>) by using (<ref>) with R=1, x'=0, t'= t/d_(x)^2. Indeed, in the case p ∈ (1,∞), we have that

    4 { t / d_(x)^2 }log v^x(x,t) =
    4 {t/d_(x)^2}log u_B ( 0, t/d_(x)^2) ≤
    4/^2 - 4/√(p') + 4 {t/d_(x)^2}log[ 
    ∫_0^π (sin)^N-p/p-1  d/∫_0^π e^-√(p')/ t (1-cos) (sin)^N-p/p-1 d].

Hence, by choosing = 2/√(p'), we conclude that

    4 t log u_B( 0, t/d_(x)^2) ≤
     -p' d_(x)^2 + 4 t log[ 
    ∫_0^π (sin)^N-p/p-1  d/∫_0^π e^-p'/2 t (1-cos)d_(x)^2 (sin)^N-p/p-1 d],

which implies (<ref>). In the case p=∞ we just need to replace (<ref>) with (<ref>) to obtain the conclusion.
 



§ BARRIERS IN THE ELLIPTIC CASE



The next two lemmas give explicit barriers for the solution u^ of (<ref>)-(<ref>) in a general domain . Compared to the parabolic case, here we obtain a sharper barrier from below, since we have a more convenient formula for the solution in the exterior of a ball.



Let ⊂^N be a domain and p∈ (1,∞]. Let u^ be the bounded (viscosity) solution of (<ref>)-(<ref>).  Pick z∈^N∖.
Then, we have that

    log{u^(x)}+√(p') {|x-z|-d_(z)}≥log e_p,z^(x)
      x∈,

where

    e_p,z^(x)=
    ∫_0^∞e^-√(p')cosh-1/|x-z|(sinh)^N-p/p-1 d/∫_0^∞e^-√(p')cosh-1/d_(z)(sinh)^N-p/p-1 d     1<p<∞,
    
    
    
    1       p=∞.




 We consider the ball B=B_R(z) with radius R=d_(z) and let v^ be the bounded solution of (<ref>)-(<ref>) relative to ^N∖B⊃. From the fact that z∈^N∖, we have that ⊂^N∖ B, which implies that

    v^≤ 1   ,

by the explicit expression of v^ given in (<ref>). Thus, by the comparison principle, we infer that v^≤ u^ on . The desired claim then follows by easy manipulations on (<ref>).



  
  Let ⊂^N be a domain and p∈ (1,∞]. Let u^ be the bounded (viscosity) solution of (<ref>)-(<ref>). 
Then, we have that

    log{u^(x)}+√(p') d_(x)≤log E_p^(d_(x)),

for every x∈, where, for ≥ 0,

    E_p^()=
    ∫_0^π(sin)^N-p/p-1 d/∫_0^πe^-√(p')1-cos/(sin)^N-p/p-1 d     1<p<∞,
    
    
    2/1+e^-2 /     p=∞.

In particular, it holds that

    log  E_p^(d_)=
    
    O( log)   1<p<∞,
    
    O()  p=∞,

as → 0^+, on every subset of  in which d_ is bounded.



For a fixed x∈, we consider the ball B^x=B_R(x) with R=d_(x) and denote by u_B^x^ the solution of (<ref>)-(<ref>) with =B^x. The comparison principle gives that

    u^≤ u_B^x^  B^x
  
and, in particular, 

    u^(x)≤ u_B^x^(x).

Observe that the uniqueness of  the solution of (<ref>)-(<ref>) and the scaling properties of _p^G imply that

    u_B^x^(x)=u_B^/R(0),

where u_B^ is the solution of (<ref>)-(<ref>) with =B, the unit ball. The explicit expressions in (<ref>) and (<ref>) then yield the pointwise estimate, since R=d_(x).
The last uniform formula then follows from (<ref>) in Theorem <ref>.







§ POINTWISE VARADHAN-TYPE FORMULAS





Set p∈ (1,∞]. Let  be a domain in ^N, with boundary  such that =(^N∖), and let u be the bounded (viscosity) solution of (<ref>)-(<ref>).
Then, we have that

    lim_t→ 0^+4tlog{u(x,t)}=-p' d_(x)^2   x∈.




It is clear that (<ref>) holds for x∈. Let x∈. We only need to apply Lemmas <ref>  and <ref>. Indeed, for z ∈^N ∖, for every and t > 0, Lemmas <ref>  and <ref> lead to

    p'( d_(x)^2 - |x - z|^2 )
     +4 t  log E^-(d_(z), t)≤
    
    4tlog{u(x,t)} + p'   d_(x)^2 ≤ 
    4 t log E^+(d_(x), t).






The last chain of inequalities implies at once that

    p'(d_(x)^2 - |x-z|^2) ≤lim inf_t→ 0^+[4tlog u(x,t) + p'  d_(x)^2]≤
    lim sup_t→ 0^+[4tlog u(x,t) + p' d_(x)^2] ≤ 0,

where we have used Lemma <ref> and the fact that tlog E^-(d_(z), t) → 0, as t→ 0^+.
Since = ( ^N ∖), we can find always a sequence of z_n that converges to a point y∈ such that d_(x)=|x-y|; by taking z=z_n in the last formula and letting n→∞, we obtain the desired claim. 

























































Let p∈ (1, ∞] and  be a domain in ^N satisfying =(^N∖); assume that u^ is the bounded (viscosity) solution of (<ref>)-(<ref>).
Then, it holds that

    lim_→ 0^+log{u^(x)}=
    -√(p') d_(x)   x∈.




 Given z∈^N∖ and >0, combining Lemmas <ref> and  <ref> gives at x∈ that

    √(p'){-|x-z|+d_(x)+d_(z)}+log e_p,z^(x)≤
    log{u^(x)} +√(p') d_(x)≤log E_p^(d_(x)).

Letting → 0^+ then gives that

    √(p'){ -|x-z|+d_(x)+d_(z)}≤
    lim inf_→ 0^+[ log{u^(x)}+√(p') d_(x)] ≤
    lim sup_→ 0^+[log{u^(x)}+√(p') d_(x)]≤
    0,

where we have used Lemma  <ref> and the fact that log e_p,z^(x) vanishes, as → 0^+. This follows by applying (<ref>) to (<ref>).
We conclude the proof as in the previous theorem, by letting z tend to y∈ such that  |x-y|=d_(x).










§ QUANTITATIVE UNIFORM FORMULAS



For an open set of class C^0, we mean that its boundary is locally the graph of a continuous function. For the sequel, it is convenient to specify the modulus of continuity, by the following definition (used in <cit.>). Let :(0,∞)→ (0,∞) be a strictly increasing continuous function such that (τ)→ 0 as τ→ 0^+. We say that an open set  is of class C^0,, if there exists a number r>0 such that, for every point x_0∈, there is a coordinate system (y',y_N)∈^N-1×, and a function :^N-1→ such that


  * B_r(x_0)∩={(y',y_N)∈ B_r(x_0):y_N<(y')};

  * B_r(x_0)∩={(y',y_N)∈ B_r(x_0):y_N=(y')}

  * |(y')-(z')|≤(|y'-z'|) for all (y',(y')), (z',(z'))∈ B_r(x_0)∩.

In the sequel, it will be useful the function defined by 

    ψ_()=min_0≤ s ≤ r√(s^2+[(s)-]^2),   ≥ 0.

This is the distance of the point z=(0',)∈^N-1× from the graph of the function .
Notice that 

    ψ()= ∈ C^k   k≥ 2

 and, otherwise, ψ()∼ C ^-1(), for some positive constant C, where ^-1 is the inverse function of . For instance, if  is of class C^, with 0<<1 — that means that  is locally a graph of an -Hölder continuous function — then ψ()≥ a ^1/ as → 0^+.

Figure <ref> describes the geometric setting considered throughout Section <ref>.





Let p ∈ (1,∞] and suppose that  is a domain of class C^0,. Let u be the bounded (viscosity) solution of (<ref>)-(<ref>).
Then, it holds that

    4t log u(x,t)+p' d_(x)^2=O(t logψ_(t))   t→ 0^+,

uniformly on every compact subset of . 
In particular, if t logψ_(t)→ 0 as t→ 0^+,
then the solution u of (<ref>)-(<ref>) satisfies (<ref>) uniformly on every compact subset of .



We need to choose z, in (<ref>), uniformly with respect to x∈. For every x∈, we fix a coordinate system (y',y_N)∈^N-1×, with its origin at a point in  at minimal distance d_(x) from x. In this coordinate system, we choose z(t)=(0',t) that, if t is small enough is by construction a point in ^N∖, since t>(0'). Also, by our assumptions on , d_(z(t)) is bounded from below by the distance of z(t) from the graph of the function y'↦(|y'|) defined for y'∈{y∈ B_r(0): y_N=0}, that is

    d_(z(t))≥min_0≤ s≤ r√(s^2+[(s)-t]^2).
 
It is clear that this construction does not depend on the particular point x∈ chosen, but only on the regularity assumptions on . 
Then, (<ref>) reads as

    p'( d_(x)^2 - |x - z(t)|^2 )
     +4 t  log E^-(d_(z(t)), t) ≤
    
    4tlog{u(x,t)} + p'   d_(x)^2 ≤ 
    4 t log
    E^+( d_(x), t),

for every x∈ and t>0.
 Observe that d_(z(t)) ≥ψ_(t) and |x-z(t)| ≤ d_(x) + |y - z(t)|. Hence, if x is such that d_(x) ≤, we have that

    -p' ( |y-z(t)|^2 + 2|y-z(t)| )
     +4 t  log E^-(ψ_(t), t) ≤
    
    4tlog{u(x,t)} + p'   d_(x)^2
    ≤ 
    4 t log E^+(, t).

From Lemma <ref>, the last term is O(t log t), as t→ 0^+.  Whereas, from the choice of z(t), the first term can be read as -p' (t^2 + 2 t) + 4 tlog E^-(ψ_(t),t) and hence its leading term is due to 4 tlog E^-(ψ_(t),t), which is a O( t logψ_(t) ), thanks to (<ref>).




































The same assertion of Theorem <ref> holds true even if we replace 1 in  (<ref>) by a bounded time-dependent non-constant  boundary data, provided that this is bounded away from zero.



Let w be the bounded solution of (<ref>), (<ref>) satisfying

    w=h  ×(0,∞),

where the function h:×(0,∞)→ is such that 

    h≤ h ≤h  ×(0,∞),
 
for some positive numbers h, h.
Then, we have that

    4 t log w(x,t)= -p'd_(x)^2+O(tlogψ_(t))   t→ 0^+,

uniformly on every compact subset of .
	

Since h  u ≤ w ≤h  u on ×(0,∞), we can apply Corollary <ref> to get:

    h  u(x,t)≤ w(x,t) ≤h  u(x,t)   ×(0,∞).

This implies that, for every x∈ and t>0,

    4t logh+4 t log u(x,t) ≤ 4 t log w(x,t) ≤  4t logh+4 t log u(x,t).

The conclusion then easily follows from Theorem <ref>.




Let p ∈ (1,∞] and  be a domain of class C^0. Suppose that u^ is the bounded (viscosity) solution of (<ref>)-(<ref>).
Then, as → 0^+, we have that 

    log{u^(x)}+ √(p') d_(x)=
    
    O()      p=∞, 
    
    O(log)      p>N.

Moreover,  if  is of class C^0,, it holds that

    log{u^(x)}+√(p') d_(x)=
      
        O(log|logψ_()|)     N=p, 
     
        O(logψ_())     1<p<N.

The formulas (<ref>) and (<ref>) hold uniformly on the compact subsets of .
In particular, if logψ_()→ 0 as → 0^+, then the convergence in (<ref>) is uniform on every compact subset of .



For any fixed compact subset K of  we let d be the positive number, defined as  

    d=
    max_x'∈ K{d_(x'), |x'|}.
 
To obtain the uniform convergence in (<ref>) we will choose z=z_ independently on x∈ K, as follows.
If  is of class C^0,, fix x∈ K, take y∈ minimizing the distance to x, and 
consider a coordinate system in ^N-1× such that y=(0', 0). If we take z_=(0',), then z_∈^N∖ when  is sufficiently small. 
With this choice, (<ref>) reads as

    √(p'){-|x-z_|+d_(x)+d_(z_)}+log e_p,z_^(x)≤
    log u^(x) +√(p') d_(x)≤log E_p^(d_(x)).

Hence, we get:

    -√(p') +log e_p,z_^(x)≤log u^(x) +√(p') d_(x)≤log E_p^(d_(x)),

since d_(z_)≥ 0 and |x-z_|≤ d_(x)+. 
Thus, if p=∞, Lemmas <ref> and (<ref>) give that

    -≤log{u^(x)}+d_(x)≤log{2/1+e^-d/},

being d_(x)≤ d, and (<ref>) follows at once.

Next, 
if 1<p<∞, we recall that log E_p^(d_(x))=O(log) on K as → 0^+, by Lemma <ref>. On the other hand, by observing that d_(z_)≥ψ(), by our assumption on , and that also |x-z_|≤ 2 d for ≤ d,
(<ref>) gives on K that

    e_p,z_^≥∫_0^∞e^-2 d √(p')/(cosh-1)(sinh)^N-p/p-1d/∫_0^∞e^-√(p')ψ()/(cosh-1)(sinh)^N-p/p-1d.

Now, after setting =N-p/2(p-1), to this formula we apply (<ref>) with =2 d √(p')/ at the numerator and (<ref>) =√(p')ψ()/ at the denominator. Thus, since the sign of  is that of N-p, on K we have as → 0  that

    log(e_p,z_^)≥ logψ()--1/2 log+O()=logψ()+O(log),

if p<N,

    log(e_p,z_^)≥ - log|logψ()|+O(log),

if p=N, and

    log(e_p,z_^)≥+1/2 log+O(),

if p>N.




Let v^:→ be the bounded solution of (<ref>) satisfying

    v^=h_  ×(0,∞),

where, for any >0, the function h_:→ is such that

    h≤ h_≤h  ,
 
for some positive numbers h, h.
Then, we have that

    log v^ (x) + √(p') d_(x) = 
     
    O()   p=∞,
    
     
    O( log)   p∈ (N,∞),
    
     
    O( log |logψ_()| )   p=N,
    
    
    O ( logψ_ () )   p∈ (1,N).

uniformly on every compact subset of .
	


The proof runs similarly to that of Corollary <ref>, having in mind Theorem <ref> instead of Theorem <ref>.











CHAPTER: ASYMPTOTICS FOR Q-MEANS



In this chapter, we mainly consider domains of class C^2. For these domains, we are able to provide further asymptotic formulas involving solutions of (<ref>)-(<ref>) and (<ref>)-(<ref>). In particular, we consider statistical nonlinear quantities, called q-means, defined, in the case of our interest, as follows. Given q∈ [1,∞], B a ball in ^N and a function u:B→, the q-mean of u on B is the unique real value μ, such that

    ‖ u-μ‖_L^q(B)≤‖ u - ‖_L^q(B),   ∈.


These quantities generalize the standard mean value, which corresponds to the case q=2. These means (there named p-means) have also been studied by Ishiwata, Magnanini and Wadade, in <cit.>, in connection with asymptotic mean value properties for p-harmonic functions.

Formulas that we give here are proper generalizations of those due to Magnanini and Sakaguchi (see <cit.> and <cit.>) concerning the linear cases. In <cit.>,  the solution of the heat equation subject to conditions (<ref>) and (<ref>) is considered and the following formula for the mean value of u on a ball touching the boundary  at only one point y_x∈ is proved:

    lim_t→ 0^+(R^2/t)^N+1/4_B_R(x)u(z,t) dz=
    C_N/√(Π_Γ(y_x)).

Here, C_N is a positive constant, x∈, R=d_(x) and { y_x}=B_R(x)∩. Also,

    Π_(y)= ∏_j=1^N-1[ 1 - R_j(y) ],   y∈,

where _1(y),…, _N-1(y) denote the principal curvatures of  at y with respect to the interior normal direction to . In <cit.>, a corresponding elliptic case is considered.
In Section <ref>, we consider the q-mean μ_q(x,t) on B_R(x) of the solution of (<ref>)-(<ref>) and obtain the formula:

    lim_t→ 0^+(R^2/t)^N+1/4(q-1)μ_q(x,t) =
     C_N,p,q {Π_Γ(y_x)}^-1/2(q-1),

This formula holds for any p∈(1,∞] and q∈(1,∞). The positive constant C_N,p,q will be specified in Theorem <ref>.
In the elliptic case, we consider the q-mean of the solution of (<ref>)-(<ref>) and, for the same values of p and q, we compute:

    lim_→ 0^+(R/)^N+1/2(q-1)μ_q,(x)=
       C_N,p,q { Π_Γ(y_x)}^-1/2(q-1),

The value of C_N,p,q can be found in Theorem <ref>.
In Theorems <ref> and <ref> also the extremal case in which q=∞ will be treated obtaining that

    lim_t→ 0^+μ_∞(x,t) = lim_→ 0^+μ_∞,(x) = 1/2.

The above limits are obtained by using improved versions of the barriers we have constructed in Chapter <ref>. These versions, that are valid for C^2-regular domains, are presented in Sections <ref> and <ref>. In Section <ref>, we first prove the asymptotic formulas for the improved barriers (see Lemmas <ref> and <ref>) and hence thanks to appropriate properties of monotonicity of the q-mean, we extend the formulas to the relevant solutions.
It is worth noting that the results of this chapter are based on Lemma <ref>, a geometrical lemma proved in <cit.>. In Section <ref>, we recall it, from <cit.>, with its complete proof.









































§ IMPROVING OF BARRIERS IN THE PARABOLIC CASE



The following lemma is a consequence of Theorem  <ref>.





Set p∈(1,∞]. 
Let  be a domain of class C^0,. Let v: ×(0,∞)→ be defined by 

    (√(p')  v(x,t)/2 √(t))= u(x,t)  (x,t)∈×(0,∞),

where u(x,t) is the (bounded) viscosity solution of (<ref>)-(<ref>).
Then, 

    v(x,t)=d_(x)+O(tlogψ_(t))   t→ 0^+,

uniformly on every compact subset of .



From the definition of v(x,t), operating as in the proof of Theorem <ref> yields that

    4t log u(x,t)+p' v(x,t)^2=4t log( √(p'/4π)∫_0^∞ e^-1/2 p' v(x,t)/√(t) -1/4 p' ^2d)≤ 0.

By this inequality, since the first summand at the left-hand side converges uniformly on every compact K⊂ as t→ 0^+, we can infer that there exist t>0 and >0 such that 0≤ v(x,t)≤ for any x∈ K and 0<t<t. 
Thus, for x∈ K we have that 

    -[4t log u(x,t)+p' d_(x)^2]+4t log( √(p'/π)∫_0^∞ e^-1/2 p' /√(t) -1/4 p' ^2d)≤
    
    p' [v(x,t)^2-d_(x)^2]≤ -[4t log u(x,t)+p' d_(x)^2],

which implies the desired uniform estimate, by means of (<ref>).


We can now refine the barriers given in Section <ref>. We define a function of t by

    η_u,K(t) = 1/√(t)max{ |v(x,t) - d_(x)| : x∈ K }  t>0.





Set p∈(1,∞]. Let  be a C^2 domain. For any compact set K⊆, we have that

    ( √(p'/4t) d_(x) + η_u,K(t) ) 
    ≤ 
    u(x,t)
    ≤( √(p'/4t) d_(x)  - η_u,K(t) )

for (x,t) ∈ K×(0,∞). It holds that η_u,K(t) = O(√(t)log t), as t→ 0^+.



We use Lemma <ref> and (<ref>) to obtain (<ref>). The asymptotic profile of η_u,K follows from (<ref>).


























§ IMPROVING OF BARRIERS IN THE ELLIPTIC CASE


To start with, we recall that a domain  of class C^2 satisfies both the uniform exterior and interior ball conditions, i.e. there exist r_i,r_e>0 such that every y∈ has the property that there exist z_i∈ and z_e∈^N∖ for which 

    B_r_i(z_i)⊂⊂^N∖B_r_e(z_e)  B_r_i(z_i)∩B_r_e(z_e)={y}.








We will also use two families of probability measures on the intervals [0, ∞) and [0,π] with densities defined, respectively, by

    dν^τ()=e^-τ (cosh-1)(sinh)^/∫_0^∞e^-τ (cosh-1)(sinh)^ d d,
           dμ^τ()=e^-τ (1-cos)(sin)^/∫_0^πe^-τ (1-cos)(sin)^ d d.



  
  Set p∈(1,∞].  Let ⊂^N be a C^2 domain. Assume that u^ is the bounded (viscosity) solution of (<ref>)-(<ref>).
If p∈ (1,∞), we set for τ_=√(p') r_e/:

    U^()=
    ∫_0^∞e^-cosh dν^τ_(),   ≥ 0,

and

    V^()=
    ∫_0^πe^-cos dμ^τ_()     0≤ < τ_, 
    
    
    {∫_0^πe^-cos dμ^0()}^-1     ≥τ_.

If p=∞, we set U^()=e^- and 

    V^()=
    cosh(τ_-)/coshτ_     0≤ < τ_, 
    
    
    
    1/cosh     ≥τ_.

Then, we have that

    U^(d_(x)//√(p'))≤ 
    u^(x)≤
    V^(d_(x)//√(p')),

for any x∈.



  Let p∈(1,∞). For any x∈ we can consider y∈ such that |x-y|=d_(x). From the assumptions on  there exists z_e∈^N∖ such that (<ref>) holds for y. As seen in the proof of Lemma <ref>, by using the comparison principle and the explicit expression (<ref>), we obtain

    u^(x)≥∫_0^∞ e^-√(p') |x-z_e|/ cosh(sinh)^ d/∫_0^∞e^-√(p') r_e/ cosh(sinh)^ d.

Thus, the fact that |x-z_e|=d_(x)+r_e gives the first inequality in (<ref>), by recalling the definition of U^.
To obtain the second inequality in (<ref>) we proceed differently whether x∈_r_i or not. Indeed, if x∈_r_i, there exists z_i∈ such that (<ref>) holds for some y∈ and x∈ B_r_i(z_i); moreover, since B_d_(x)(x)∩ B_r_i(z_i)={y}, we observe that x lies in the segment joining y to z_i, and hence |x-z_i|=r_i-d_(x). Again, by using the comparison principle and the expression in (<ref>), we get that 

    u^(x)≤∫_0^πe^√(p') cos |x-z_i|/(sin)^ d/∫_0^πe^√(p') cos r_i/(sin)^ d,

that, by using the definition of V^ and the fact that |x-z_i|=r_i-d_(x), leads to the second inequality in (<ref>).
If x∈∖_r_i, we just note that the expression of V^ was already obtained in Lemma <ref>. 
The case p=∞ can be treated with similar arguments. 













































































































§ ASYMPTOTICS FOR Q-MEANS















Before going on, we recall from <cit.> some preliminary facts about q-means and a geometrical lemma from <cit.>.

For any continuous function u there exists an unique μ satisfying (<ref>) (see <cit.>). For 1≤ q < ∞, μ can be characterized by the equation

    ∫_B | u(z) - μ |^q-2[ u(z) - μ]  dz =0.

This equation is equivalent to 

    ∫_B [ u(z) - μ]_+^q-1 dz = ∫_B [μ - u(z) ]_+^q-1 dz,

where [s]_+=max{0,s}, for s∈.
The q-mean is monotonically increasing with respect to u, in the sense that 

    μ^u ≤μ^v   u ≤ v   B.

where, μ^u and μ^v are respectively the q-mean of u and of v.



The next lemma is a version of <cit.> slightly adapted to our notations. For the reader's convenience, we also report its proof. We recall that by Π_ we mean the function in (<ref>).



Let x∈ and assume that, for R>0, there exists y_x∈ such that B_R(x)∩(^N∖)={y_x} and that _j(y_x)<1/R for j=1,…,N-1. Set _s={y∈:d_(y)=s}, for s>0.
Then, it holds that

    lim_s→ 0^+s^-N-1/2_N-1(_s∩ B_R(x))=_N-1 (2R)^N-1/2/(N-1)√(Π_(y_x)),

where _N-1 denotes (N-1)-dimensional Hausdorff measure and _N-1 is the surface area of a unit sphere in ^N-1.



Without loss of generality, we can suppose that we are working with a coordinate system {z_1,…,z_N}, such that y_x=0, the tangent plane to  at y_x coincides with the plane {z_N=0} and x=(0,…,0,R). We can also suppose that z_1,…,z_N-1 are chosen such that

    d_(z)= z_N - 1/2∑_j=1^N-1_j(y_x) z_j^2 + o(|z|^2)
    
     d_/ z_N(z)=1 + o(|z|).

Note that, with these choices, B_R(x) is represented by the inequality |z'|^2+(z_N-R)^2 < R^2, where z'=(z_1,…,z_N-1).
Hence, near the origin, B_R(x) is represented by

    z_N = 1/2|z'|^2+O(|z'|^3).

Combining (<ref>) with (<ref>), gives

    d_(z)= 1/2∑_j=1^N-1(1/R - _j(y_x)) z_j^2 + o(|z'|^2)  z∈ B_R ∩ B_R(x).

Since B_R(x)∩(^N∖)={0}, for every > 0, there exists s_>0 such that

    _s ∩ B_R(x) ⊆ B_  0<s<s_.

Hence, from (<ref>), if > 0 is sufficiently small and 0<s<s_, _s ∩ B_R(x) is represented by the graph of a smooth function z_N=ψ(z'). Differentiating d_(z',ψ(z'))=s with respect to z_j yields

    d_z_N_z'ψ + _z' d = 0,

which together with | d_|=1 implies that

    √(1 + |_z'ψ|^2) =  1/d_z_N.

Projecting _s∩ B_R(x) orthogonally on {z_N = 0} yields a domain A_s⊆^N-1. Let η > 0 be sufficiently small. From (<ref>) and (<ref>), there exists _0>0 such that, for every 0<s<s__0, we have

    E_s^+ ⊆ A_s ⊆ E_s^-

where 

    E_s^± = { z'∈^N-1: 1/2∑_j=1^N-1(1/R - _j(y_x) ±η) z_j^2 < s}.

Moreover, combining (<ref>) and (<ref>) yields

    1 ≤√(1 + |_z'ψ|^2)≤ 1 + η,

for very 0<s<s__0. Hence, it follows from  (<ref>) and (<ref>) that

    ∫_E_s^+1  dz' ≤_N-1(_s∩ B_R(x)) ≤∫_E_s^-(1+η) dz',

for every 0<s<s__0, since

    _N-1(_s∩ B_R(x) ) = ∫_A_s√(1 + |_z'ψ|^2) dz'.

Hence, from (<ref>) we see that

    _N-1 2^N-1/2/(N-1){∏_j=1^N-1[1/R - _j(y_x) +η] }^-1/2≤
    
    s^-N-1/2_N-1(_s∩ B_R(x) )
    ≤
    _N-1 2^N-1/2/(N-1){∏_j=1^N-1[1/R - _j(y_x) -η] }^-1/2

for every 0<s<s__0. Since η > 0 is arbitrarily small, we conclude the proof.




 §.§ Short-time asymptotics for q-means
















Set 1<q<∞, let x∈, and assume that, for R>0, there exists a point y_x∈ such that B_R(x)∩(^N∖)={y_x} and _j(y_x)<1/R for j=1,…,N-1.
Let ξ, η:(0,∞)→(0,∞) be two functions of time such that 
ξ(t) is positive in (0,∞), and

    lim_t→ 0^+ξ(t)=lim_t→ 0^+η(t)=0.
 
For a non-negative, decreasing and continuous function f on  such that

    ∫_0^∞ f()^q-1^N-1/2 d<∞,

set

    w(z,t)=f(d_(z)/ξ(t)+η(t))   (z,t)∈×(0,∞).

If μ_q^w(x,t) is the q-mean of w(·, t) on B_R(x), 
then the following formula holds:

    lim_t→ 0^+(R/ξ(t))^N+1/2(q-1)μ_q^w(x,t)=
    {2^-N+1/2N! ∫_0^∞ f()^q-1^N-1/2 d/( N+1/2)^2√(Π_(y_x)) }^1/q-1 .




We know from (<ref>) that μ(t)=μ_q^w(x,t) is the unique root of the following equation

    ∫_B_R(x)[w(z,t)-μ(t)]_+^q-1 dz=∫_B_R(x)[μ(t)-w(z,t)]_+^q-1 dz.

Firstly, we compute the short-time behavior of the left-hand side of (<ref>). Let _s={z∈ B_R(x):d_(z)=s}. By the co-area formula, we get that

    ∫_B_R(x)[w(z,t)-μ(t)]_+^q-1 dz=
    ∫_0^2R[f(s/ξ(t)+η(t))-μ(t)]^q-1_+_N-1(_s) ds.

By the change of variable s=ξ(t)[-η(t)], we obtain that

    ∫_B_R(x)[w(z,t)-μ(t)]_+^q-1 dz=
    ξ(t)∫_η(t)^(t)[f()-μ(t)]^q-1_+_N-1(_ξ(t)[-η(t)]) d,

where we set (t)=2R/ξ(t)+η(t). 
Hence,

    ξ(t)^-N-1/2∫_B_R(x)[w(z,t)-μ(t)]_+^q-1 dz=
      
    ∫_η(t)^(t)_N-1(_ξ(t)[-η(t)])/{ξ(t)[-η(t)]}^N-1/2[-η(t)]^N-1/2{f()-μ(t)}^q-1 d.

Now, as t→ 0^+ we have that η(t), ξ(t), μ(t)→ 0, (t)→∞ and that ξ(t)[-η(t)]→ 0 for almost every ≥ 0. Thus, we can infer that

    lim_t→ 0^+ξ(t)∫_B_R(x)[w(z,t)-μ(t)]_+^q-1 dz=
    _N-1(2R)^N-1/2/(N-1)√(Π_(y_x))∫_0^∞f()^q-1^N-1/2 d,








by Lemma <ref> and an application of the dominated convergence theorem, as an inspection of the integrand function reveals.
Secondly, we treat the short-time behavior of the right-hand side of (<ref>). By again performing the co-area formula and after some manipulations, we have that

    ∫_B_R(x) [μ(t)-w(z,t)]^q-1_+ dz=
    
    μ(t)^q-1∫_0^2R[1-f(s/ξ(t)+η(t))/μ(t)]^q-1_+_N-1(_s) ds

which, on one hand,  leads to 

    ∫_B_R(x) [μ(t)-w(z,t)]^q-1_+ dz≤μ(t)^q-1|B_R(x)|.

Notice in particular that, by using both (<ref>) and (<ref>), the last inequality informs us that

    μ(t)≥ c ξ(t)^N+1/2(q-1),

for some positive constant c. Hence, after setting (s,t)=s/ξ(t)+η(t), the assumptions on f give the following chain of inequalities:

    ∫_(s,t)/2^∞f()^q-1^N-1/2 d≥∫_(s,t)/2^(s,t)f()^q-1^N-1/2 d≥
    2(1-2^-N+1/2)/N+1f((s,t))^q-1/ξ(t)^N+1/2[s+η(t) ξ(t)]^N+1/2≥
    2(1-2^-N+1/2)/c(N+1)[f(s/ξ(t)+η(t))/μ(t)]^q-1[s+η(t) ξ(t)]^N+1/2.

Since, for almost every s≥ 0, the first term of the chain vanishes as t→ 0^+, we have that

    lim_t→ 0^+f(s/ξ(t)+η(t))/μ(t)=0,

for almost every s≥ 0. Thus, (<ref>) gives at once that

    lim_t→ 0^+μ(t)^1-q∫_B_R(x) [μ(t)-w(z,t)]^q-1_+ dz=|B_R(x)|.

Finally, (<ref>), (<ref>) and (<ref>) tell us that

    μ(t)^q-1=ξ(t)^N+1/2_N-1 (2R)^N-1/2/(N-1)√(Π_(y_x)) ∫_0^∞ f()^q-1 ^N-1/2 d+o(1)/|B_R(x)|+o(1),

that gives (<ref>), after straightforward calculations involving Euler's gamma function.



If q=∞, we know that

    μ_∞^w(x,t)=1/2 {min_B_R(x) w(·,t)+max_B_R(x) w(·,t)}=
    1/2 [f(d/ξ(t)+η(t))+f(η(t))],

where d is positive, being the maximum of d_ on B_R(x). Hence, it is easy to compute:

    lim_t→ 0^+μ_∞^w(x,t)=1/2 f(0).

Thus, formula (<ref>) does not extend continuously to the case q=∞.





Let x∈, and assume that, for R>0, there exists a point y_x∈ such that B_R(x)∩(^N∖)={y_x} and _j(y_x)<1/R for j=1,…,N-1.
Set 1<p≤∞ and suppose that u is the bounded (viscosity) solution of (<ref>)-(<ref>) and, for 1<q≤∞, μ_q(x,t) is the q-mean of u(·, t) on B_R(x).
Then, if 1<q<∞, the following formulas hold:







    lim_t→ 0^+(R^2/t)^N+1/4(q-1)μ_q(x,t)=
    
    {N! ∫_0^∞()^q-1^N-1/2 d/ ( N+1/2)^2}^1/q-1{ p'^N+1/2 Π_Γ(y_x)}^-1/2(q-1),

and

    lim_t→ 0^+μ_∞(x,t)=1/2.




By using (<ref>) and (<ref>), the limit in (<ref>) will result from Lemma <ref>, where we choose:

    w(x,t)=(√(p'/4t) d_(y)±η(t)),

that is we choose ξ(t)=√(4t/p') and η(t) is given by (<ref>), with K=B_R(x).
Thus, (<ref>) will follow at once from (<ref>), where
f()=().
By the same argument, we also get the case q=∞, since f(0)=1.
 


Notice that

    {∫_0^∞()^q-1^N-1/2 d}^1/q-1

can be seen as the (q-1)-norm of  in (0,∞) with respect to the weighed measure ^N-1/2 d.





 §.§ Asymptotics for q-means in the elliptic case


















The next lemma gives the asymptotic formula for → 0^+ for the q-mean on B_R(x) of a quite general class of functions, which includes both the barriers U^ and V^ of Lemma <ref>.

 

Set 1<q<∞. Let x∈ and assume that, for R>0, there exists y_x∈ such that B_R(x)∩(^N∖)={y_x} and that _j(y_x)<1/R for j=1,…,N-1.
Let {ξ_n}_n∈ and {f_n}_n∈ be sequences such that


  * ξ_n>0 and ξ_n→ 0 as n→∞;

  * f_n:[0,∞)→[0,∞) are decreasing functions;

  * f_n converges to a function f almost everywhere as n→∞;

  * it holds that

    lim_n→∞∫_0^∞f_n()^q-1 ^N-1/2 d=
    ∫_0^∞f()^q-1 ^N-1/2 d,

and the last integral converges.

For some 1<q<∞, let μ_q,n(x) be the q-mean of f_n(d_/ξ_n) on B_R(x).
Then we have:

    lim_n→∞(R/ξ_n)^N+1/2(q-1)μ_q,n(x)=
    {2^-N+1/2N! ∫_0^∞f()^q-1^N-1/2 d/(N+1/2)^2 √(Π_(y_x) )}^1/q-1.




  From (<ref>), we know that μ_n=μ_q,n(x) is the only root of the equation
  
    ∫_B_R(x)[f_n(d_/ξ_n)-μ_n]_+^q-1 dz=
    ∫_B_R(x)[μ_n-f_n(d_/ξ_n)]_+^q-1 dz,

where we mean [t]_+=max(0, t).
Thus, if we set

    _={y∈ B_R:  d_(y)=},

by the co-area formula 
we get that

    ∫_B_R(x)[f_n(d_/ξ_n)-μ_n]_+^q-1 dz=
    ∫_0^2R[f_n(/ξ_n)-μ_n]_+^q-1_N-1(_) d,

that, after the change of variable =ξ_n τ and easy manipulations, leads to the formula:

    ∫_B_R(x)[f_n(d_/ξ_n)-μ_n]_+^q-1 dz=
    ξ_n^N+1/2∫_0^2R/ξ_n[f_n(τ)-μ_n]_+^q-1τ^N-1/2[_N-1(_ξ_nτ)/(ξ_nτ)^N-1/2] dτ.

Therefore, since μ_n→ 0 as n→∞, an inspection of the integrand at the right-hand side, assumptions (i)-(iv), and Lemma <ref> make it clear that we can apply the generalized dominated convergence theorem (see <cit.>) to infer that

    lim_n→∞ξ_n^-N+1/2∫_B_R(x)[f_n(d_/ξ_n)-μ_n]_+^q-1 dz=
    (2R)^N-1/2_N-1/(N-1)√(Π_(y_x))∫_0^∞f()^q-1^N-1/2 d.







Next, by employing again the co-area formula, the right-hand side of (<ref>) can be re-arranged as

    ∫_B_R(x) [μ_n-f_n(d_/ξ_n)]_+^q-1 dz=
    μ_n^q-1∫_0^2R[1-f_n(/ξ_n)/μ_n]_+^q-1_N-1(_) d,

that leads to the formula

    lim_→ 0^+μ_n^1-q∫_B_R(x) [μ_n-f_n(d_/ξ_n)]_+^q-1 dz=
    |B_R|,

by dominated convergence theorem, if we can prove that

    f_n(/ξ_n)/μ_n→ 0   n→∞,

for almost every ≥ 0.
Then, after straightforward computations, (<ref>) will follow 
by putting together (<ref>), (<ref>) and (<ref>).
We now complete the proof by proving that (<ref>) holds. From (<ref>), (<ref>), and the fact that

    ∫_B_R(x)[μ_n-f_n(d_/ξ_n)]_+^q-1 dz≤μ_n^q-1 |B_R|,

we have that there is a positive constant c such that

    μ_n^1-q≤ c ξ_n^-N+1/2.

Also, for every τ>0 we have that

    ∫_τ/2ξ_n^∞ f_n()^q-1 ^N-1/2 d≥∫_τ/2ξ_n^τ/ξ_n f_n()^q-1 ^N-1/2 d≥
    2(1-2^-N+1/2)/N+1 f_n(τ/ξ_n)^q-1(τ/ξ_n)^N+1/2≥
    2(1-2^-N+1/2)/c (N+1) τ^N+1/2{f_n(τ/ξ_n)/μ_n}^q-1.

Thus, (<ref>) follows, since the first term of this chain of inequalities converges to zero as n→∞, under our assumptions on f_n and ξ_n, in virtue of the generalized dominated convergence theorem.



The case q=∞ is simpler. From <cit.> and then the monotonicity of f_n we obtain that:

    μ_∞,n(x)=1/2{min_B_R(x)f_n(d_/ξ_n)+max_B_R(x)f_n(d_/ξ_n)}= 
    1/2{f_n(2R/ξ_n)+f_n(0)}.

Thus, if we replace the assumptions (iii) and (iv) by f_n(0)→ f(0) as n→∞, we conclude that
μ_∞,n(x)→ f(0)/2, since f_n(2R/ξ_n)→ 0 as n→∞. 




  
Set 1<p≤∞. Let x∈ be such that B_R(x)⊂ and B_R(x)∩(^N∖)={y_x}; suppose that k_j(y_x)<1/R, for every j=1,…,N-1.
Let u^ be the bounded (viscosity) solution of (<ref>)-(<ref>) and, for 1<q≤∞, let μ_q,(x) be the q-mean of u^ on B_R(x).
Then, if 1<q<∞, we have that 

    lim_→ 0^+(/R)^-N+1/2(q-1)μ_q,(x)=
      {2^-N+1/2N!/(q-1)^N+1/2(N+1/2)}^1/q-1{p'^N+1/2Π_(y_x)}^-1/2(q-1).

If q=∞, we simply have that 

    lim_→ 0^+μ_∞,(x) =1/2.




We have that μ_q,^U^(x)≤μ_q,(x)≤μ_q,^V^(x) by the monotonicity of the q-means, where with μ_q,^U_ and μ_q,^V^ we denote the q-mean of U^(d/) and V^(d/) on B_R(x).
Hence, in order to prove (<ref>), we only need to apply Lemma <ref> to  f_n=U^_n and f_n=V^'_n, where the vanishing sequences _n and '_n are chosen so that the lim inf and lim sup of (/R)^-N+1/2(q-1)μ_q,(x)
as → 0 are attained along them, respectively.
By an inspection, it is not difficult to check that f_n=U^_n and f_n=V^'_n, with ξ_=/√(p') and f()=e^-, satisfy the relevant assumptions of Lemma <ref>, by applying, in particular, Lemma <ref> for (iii) and the dominated convergence theorem for (iv).
 









CHAPTER: GEOMETRIC AND SYMMETRY RESULTS



The goal of this chapter is to collect some geometric and symmetry results for solutions (<ref>)-(<ref>) or (<ref>)-(<ref>), in the spirit of those given by Magnanini and Sakaguchi in <cit.>. We obtain characterizations of balls, spheres and hyperplanes as applications of Varadhan-type formulas of Chapter <ref> and of formulas for q-means of Chapter <ref>. 
We introduce the problems that we consider. We say that an (N-1)-dimensional surface  is a time-invariant level surface for the solution u of (<ref>)-(<ref>), if there exists a function a_:(0,∞)→ (0,∞) such that

    u(x,t)=a_(t)   (x,t)∈×(0,∞).



In the case of the heat equation, a time-invariant level surface is commonly called stationary isothermic surface.  We list two results concerning stationary isothermic surfaces, which are relevant for the analysis carried out in this chapter. In <cit.>, for a bounded domain  that satisfies  the exterior sphere condition, if (<ref>) holds for = D, where D satisfies the interior cone condition and  D⊂, then  must be a ball. See also <cit.> and <cit.>. In <cit.>, the case of domains with non-compact boundary is considered. In fact in <cit.>, the authors have shown that if  is the graph of function defined on the whole ^N-1, satisfying certain sufficient assumptions, and (<ref>) holds, then  must be a hyperplane. See also <cit.>.



In Section <ref>, by employing the method of moving planes (see <cit.>) as in <cit.> and <cit.>, we give a proper version of <cit.>  in the case p∈(1,∞). See Corollary <ref>. This corollary is actually a consequence of a quite more general theorem in which one obtains the spherical symmetry under the weaker condition that there exist t>0 and R>0 such that

    x↦ u(x,t)  _R,

where _R = { x∈:d_(x)=R}.
See Theorem <ref>.



In the case (<ref>)-(<ref>) we consider surfaces  that are level sets of u^, for any > 0, i.e.  satisfies the requirement:

    u^  ,   > 0.

We point out that, in the linear case, from (<ref>)



 it follows that  is a stationary isothermic surface if and only if (<ref>) holds true.
 In Section <ref>, we present results which are analogous to those obtained for p 2 in the parabolic case, by just adapting the proofs in the elliptic context. 

Section <ref> contains our result for the case of non-compact boundaries. There, we generalize  the result in <cit.> to a generic p∈(1,∞) and to the elliptic case.
Section <ref> contains another type of symmetry result for the solution of (<ref>)-(<ref>). It concerns the following condition for the q-mean μ_q(x,t) of a ball B_R(x) such that R=d_(x). Let  be a domain with bounded and connected boundary in which there exists a parallel surface _R, such that

    x↦μ_q(x,t)   _R,

for any fixed t>0. In the spirit of <cit.>, in Theorem <ref> we show that if (<ref>) holds, then  must be a sphere. Theorem <ref> takes care of the elliptic counterpart of (<ref>).





























































§ PARALLEL SURFACES



An interesting geometric property that invariant surfaces enjoy is that they are parallel to , as the following results show for both the parabolic and elliptic case.



Let  be a domain in ^N satisfying =(^N∖) and suppose that, for 1<p≤∞, u is the solution of (<ref>)-(<ref>). 
If ⊂ is a time-invariant level surface for u, then there exists R>0 such that 

    d_(x)=R  x∈.




Let R=(,) and let x_0 be a point in  such that d_(x_0) = R. If y∈, we have that u(x_0,t)=u(y,t) and hence 4t log u(x_0,t)=4t log u(y,t) for every t>0. By Theorem <ref>,
we infer that d_(x_0)=d_(y) and hence we obtain our claim.




Let  be a domain in ^N satisfying =(^N∖) and suppose that, for 1<p≤∞, u^ is the solution of (<ref>)-(<ref>). 
If ⊂ is an level surface of u^, for every > 0, then there exists R>0 such that (<ref>) holds true.






To conclude, it is sufficient to properly modify the proof of Theorem <ref>, having in mind Theorem <ref>.




§ SPHERICAL SYMMETRY FOR INVARIANT LEVEL SURFACES



We recall that bounded viscosity solutions of (<ref>) and (<ref>) are of class C^1,_loc, for some 0 <  < 1. See <cit.> and <cit.>. 




In Theorems <ref> and <ref>, we apply the method of moving planes to a subset D of a bounded domain . The idea is to show that D is mirror symmetric in every direction. We introduce some notations.

Given a direction ∈^N-1 and ∈, let π_={x∈^N: x,  = }. For a fixed point x∈^N, we define x^*, the reflection of x in π_, by 

    x^*= x -2(  x, - ).

Let H_ = {x∈^N:  x,  > } so that π_ =  H_. Also, let D_ = D ∩ H_ and D^*_ its reflection in π_. We set = sup{ x,  : x∈ D}. Suppose that D is of class C^2. From <cit.>, for < sufficiently close to , we have that D^*_⊆ D. Here, we observe that, from <cit.>, if D^*_⊂ D, then _^* ⊆.
Let ^* be the number defined by

    ^*=inf{ < : D_μ^* ⊆ D:  < μ < }.

Eventually, one of the following two cases occurs:


  * the boundary of D^*_^* becomes tangent to that of D, and the set of tangency contains points not belonging to π_^*.

  * π_^* is orthogonal to the boundary of D and of D^*_^*, at some point of the intersection.




Set p∈(1,∞). Let  be a bounded domain of class C^2 and let u be the solution of (<ref>)-(<ref>). Suppose that D is a C^2 domain such that D⊂, D=_R, for some R>0, and there exists t>0 for which u is constant on _R×{t}.






Then,
D and
 must be concentric balls. 












Since _^*^*⊂, we can define the function u^*:^*_^*×(0,∞)→, by u^*(x,t)=u(x^*,t), for x∈_^*^* and t>0. It is easy to see that

    u^*_t-_p^G u^*=0      ^*_^*×(0,∞),
    
    u^*=0      ^*_^*×{0},
    
    u^*≥ u      ^*_^*×(0,∞).

Moreover, by the (weak) comparison principle (see Corollary <ref>), we have that the function w=u - u^* is non-positive on the whole ^*_^*×(0,∞).
Now, following the proof of <cit.> (see also <cit.> <cit.>), we can apply the strong maximum principle to w, in a proper sub-cylinder of ×(0,∞). 
Consider v(x)=u(x,t), for x∈ D. From Lemma <ref>, we have that v is a non-constant viscosity subsolution of -_p^G v = 0 in D. Moreover, v equals a constant on _R. Thus, by applying the strong maximum principle (see Remark <ref>) and Corollary <ref>, we obtain that 

    v≠ 0  _R,

where we use that u is differentiable in ×(0,∞).
Also, we have that there exist > 0 and t_1, t_2 > 0, with t_1 < t < t_2,  such that 

    u≠ 0  S_× (t_1,t_2),

where S_={y∈:d__R(y)< }. Observe that the last set displayed in (<ref>) is a neighborhood of _R. 

Hence, by using (<ref>), we have that | u|,| u^*|>0 in (S_∩_^*^*)×(t_1,t_2). Therefore, by a standard procedure, we can infer that w is a (non-positive) solution of the following uniformly parabolic equation with smooth coefficients
 
    w_t-{A'^2 w} - b · w =0   (S_∩_^*^*)×(t_1,t_2),

where A' and b are defined, respectively, by the following expressions:

    A'=∫_0^1_X F( u + (1- ) u^*, ^2 u+(1-)^2 u^*)d

and

    b=∫_0^1_ξ F( u + (1- ) u^*, ^2 u+(1-)^2 u^*)d.

Here, F is defined in (<ref>). Observe that (as shown in <cit.>) the matrix A' is uniformly elliptic and the coefficients b are bounded in (S_∩_^*^*)×(t_1,t_2).
Applying the classical strong maximum principle to (<ref>) (see <cit.>) yields that either w ≡ 0 on (S_∩^*_^*)× [t_1,t_2] or w<0 in (S_∩_^*^*)×(t_1,t_2).
Now, we conclude as in <cit.>. Suppose that D^*_^*⊂ D∖D_^*. Then S_∩_^*^* contains points that are in D_^*^*∖ D. This implies that

    w<0   (S_∩_^*^* )×(t_1,t_2)
 
and that

    w/ > 0   (S_∩π_^*) × (t_1,t_2).








If (1) occurs, then there exists y ∈ D_^*^* ∩ D. This implies that u(y,t)= u^*(y,t) since D is a level surface of u(·, t) and hence that w(y,t)=0 which contradicts (<ref>). 
If (2) occurs, then D must be orthogonal to π_^* at some z ∈ D, then u/(z,t) = 0. Also, we have that u^*/(z,t) = 0 and then that w/ (z,t) = 0. This contradicts (<ref>), since w/(z,t) > 0, by (<ref>).
Hence, D must be symmetric with respect to every direction and hence D must be a ball. Since  and D are C^2, then we conclude that  must be a ball.




We point out that in Theorem <ref>, we can replace the C^2 regularity of _R with a weaker assumption. Indeed, we can apply the Hopf-Oleinik lemma to infer that u ≠ 0 on _R by assuming that _R admits an interior -pseudo ball condition, where  satisfies the assumptions of Lemma <ref>. For example, if we assume _R ∈ C^1,, for some ∈(0,1), then Theorem <ref> still holds true.



Now, we are ready to obtain the following characterization of balls, as a consequence of Theorem <ref>.



Set p∈(1,∞). Let  be a bounded domain of class C^2 and  u be the solution of (<ref>)-(<ref>). Suppose that D is a C^2 domain such that D⊂ and with boundary D= satisfying (<ref>).
Then, D and  must be concentric balls.



It is enough to apply Theorem <ref>. Indeed, we just note that, from Theorem <ref>, there exists R>0, such that =_R.



With some adjustments, we obtain the same conclusions of Theorem <ref> and Corollary <ref> in the case of the elliptic problem (<ref>)-(<ref>). 



Set p∈(1,∞). Let  be a bounded domain of class C^2 and u^ be the solution of (<ref>)-(<ref>). Suppose that  D is a C^2 domain such that D⊂, D = _R, for some R>0, and there exists >0 for which u=u^ is constant on _R.

Then, D and  must be concentric balls.



It is well defined the function u^*:_^*^*→, by u^*(x)=u(x^*), for x∈_^*^*. Then, it is an easy check to prove that

    u^*-^2 _p^G u^*=0      _^*^*,
    
    u^*≥ u      ^*_^*.

Hence, by applying the comparison principle (see Corollary <ref>), we have that w=u - u^*≤ 0 on _^*^*. 
Now, proceeding as in the proof of Theorem <ref>, by applying Corollary <ref> and <ref>, we have that | u|, | u^*|>0 in S_, the set  defined in the proof of Theorem <ref>. By using standard elliptic regularity theory, we have that w is a (non-positive) solution of a smooth uniformly elliptic equation

    w-^2 {A' ^2 w} - ^2 b,  w  =0   S_,

where A' and b have the same structure of those of the proof of Theorem <ref>. Since w≤ 0, we can apply to w the classical strong maximum principle (see <cit.>), which implies that either w≡ 0 on S_∩^*_^* or w<0 in S_∩_^*^*. Hence, we conclude as in the proof of Theorem <ref>.




Set p∈(1,∞). Let  be a bounded domain of class C^2 and u^ be the solution of (<ref>)-(<ref>). Suppose that D is a domain of class C^2, such that D⊂. Suppose that D is a level surface of u^, for any >0.
Then, D and  must be concentric balls.



We conclude from Theorems <ref> and <ref>.




































































































































































































































§ THE CASE OF NON-COMPACT BOUNDARIES










Let f:^N-1→ be a continuous function. In this Subsection, we consider domains of the following form

    ={(x',x_N)∈^N-1×:x_N>f(x')}.




Set p∈(1,∞). Let  be defined by (<ref>), with f of class C^2. Suppose that there exists a basis {ξ^1,…,ξ^N-1}⊂^N-1 such that  for every j=1,…, N-1, the function f(x'+ξ^j)-f(x') has either a maximum or a minimum in ^N-1.  Let u be the bounded solution of (<ref>)-(<ref>).  Suppose that there exists R>0 such that _R is of class C^2 and t>0 for which one of the following occurs:


  * u(·,t) is constant on _R.

  * for some q∈[1,∞), the function x↦μ_q(x,t) is constant on _R.

Then, f is affine and so  is a hyperplane.



From the assumption on f, for a fixed j=1,…,N-1, the function of x' defined by f(x'+ξ^j)-f(x') has either a maximum or a minimum in ^N-1. Say f(x'+ξ^j)-f(x') has a maximum M in ^N-1. Then there exists z' ∈^N-1 such that 

    f(x'+ξ^j)-f(x') ≤ M = f(z'+ξ^j) - f(z')  x'∈^N-1.

We apply the sliding method, a variant of the method of moving planes, introduced in <cit.>. Here, we adapt the proof given with that method in <cit.>. Also see <cit.>. Set 

    _ξ^j,M={(x',x_N)∈^N: (x'+ξ^j, x_N + M)∈}.

We have that ⊆_ξ^j,M and that z=(z',f(z')) ∈∩()_ξ^j,M. In particular, from the regularity of both  and _R we can infer that, if ν_ represents the unit outward normal to , then y=z - ν_(z) R  ∈_R ∩(_R)_ξ^j,M, which means that y + (ξ^j,M)∈_R. 
Since ⊆_ξ^j,M, we can define the function u^*:×(0,∞)→, by u^*(x,t)=u(x+(ξ^j,M),t), for x∈ and t>0. It easy to see that 

    u^*_t-_p^G u^*=0      ×(0,∞),
    
    u^*=0      ×{0},
    
    u^*≤ u      ×(0,∞).

Thus, from the comparison principle (Corollary <ref>), u^*≤ u on ×(0,∞). 
Now, suppose that ⊂_ξ^j, M. As we have done in the proof of Theorem <ref>, we show that we can use the strong comparison, in a proper sub-cylinder of ×(0,∞). By using Corollaries <ref> and <ref> and the continuity of u, there exist >0 and t_1 < t < t_2 such that

    u≠ 0  S_× (t_1,t_2),

where here S_ is the compact set B_R(y) ∩(_R + B_).
Thus, u^*-u is a solution in S_× (t_1,t_2) of the uniformly parabolic equation with smooth coefficients (<ref>) and hence we can apply to u^* - u the strong maximum principle. In particular, since  does not coincide with _ξ^j,M, we infer that

    u^*<u   S_×(t_1,t_2).
 
Denoting with μ^*_q(y,t) the q-mean of u^*(·,t) on B_R(y), the last inequality implies that 

    μ_q^*(y,t)<μ_q(y,t),

since |{u^* < u}∩ B_R(y)| > 0. Indeed, from (<ref>), we have that μ^* ≤μ and from the fact that the function u↦ |u-μ^*|^q-2 (u-μ^*) is strictly increasing, we have that





    0= ∫_B_R(y) |u^*(ζ,t) - μ^*|^q-2(u^*(ζ,t) - μ^*)  dζ <
    ∫_B_R(y) |u(ζ,t) - μ^*|^q-2 (u(ζ,t) - μ^*) dζ

which implies that μ^* ≠μ.
Now, we prove that if either (i) or (ii) holds, we find a contradiction to (<ref>) or (<ref>). If (i) holds then we have that u(y,t) = u(y+(ξ^j,M), t) = u^*(y,t), which contradicts (<ref>) at once. If (ii) holds, then

    μ_q(y,t)=μ_q(y + (ξ^j,M),t)=μ_q^*(y,t),

where the first equality is due to (ii) and the latter one  is based on the following argument. Employing a change of variable ζ = ζ' + (ξ^j,M) yields that





    ∫_B_R(y + (ξ^j,M)) |u(ζ , t)-|^q-2(u(ζ,t)-)  dζ =
    ∫_B_R(y) |u^*(ζ',t)-|^q-2(u^*(ζ', t)-)  dζ',

for any ∈, which implies the latter equation in (<ref>), by the definition of q-means. Hence, we have found again a contradiction. Hence, we must have _ξ^j,M=.
Now, we conclude as in <cit.>. Indeed, we have that for any 1≤ j ≤ N-1, for any x'∈^N-1,

    f(x'+ξ^j)-f(x') = a_j,

for some a_j∈. The continuity of f, the fact that {ξ^1,…, ξ^N-1} is a basis of ^N-1 and an iteration of the sliding method imply that

    f(x'+y')-f(x')=f(z'+y')-f(z')   x',y',z' ∈^N-1.

Since f is continuous, solving the latter system of functional equations yields that f must be affine.












Set p∈ (1,∞). Let  be defined by (<ref>), with f of class C^2. Suppose that there exists a basis {ξ^1,…,ξ^N-1}⊂^N-1 such that  for every j=1,…, N-1, the function f(x'+ξ^j)-f(x') has either a maximum or a minimum in ^N-1. Let u be the bounded solution of (<ref>)-(<ref>). Suppose that there exists a surface  of class C^2 such that (<ref>) holds true.
Then f is affine and  is a hyperplane.



It is sufficient to observe that, from Theorem <ref>, there exists R>0 such that =_R. Hence, we apply Theorem <ref>.


Also in this case, we can give the corresponding theorem in the case (<ref>)-(<ref>).



Set p∈(1,∞). Let  be defined by (<ref>), with f of class C^2. Suppose that there exists a basis {ξ^1,…,ξ^N-1}⊂^N-1 such that  for every j=1,…, N-1, the function f(x'+ξ^j)-f(x') has either a maximum or a minimum in ^N-1. Let u^ be the bounded solution of (<ref>)-(<ref>). Suppose that there exists R>0 such that _R is of class C^2 and >0 for which one of the following occurs:


  * u^ is constant on _R.

  * for some q∈[1,∞), the function x↦μ_q,(x) is constant on _R.

Then, f is affine and so  is a hyperplane.



With the same notations as in the proof of Theorem <ref>, we have that, for j=1, …, N-1, we can define the function u^*:→, by u^*(x)=u^(x+(ξ^j,M)). For the comparison principle and the maximum principle (Corollaries <ref> and <ref>), we have that u^*-u^≤ 0, on . 
Moreover, there exists >0 such that | u^*|, | u|>0 in S_. Thus, u^*-u^ satisfies the uniformly elliptic equation with smooth coefficients (<ref>) in S_.
Supposing that ⊂_ξ^j,M yields that u^*-u^<0, in S_, from the strong maximum principle. Thus, we conclude as in the proof of Theorem <ref>.




Set p∈(1,∞). Let  be defined by (<ref>), with f∈ C^2. Suppose that there exists a basis {ξ^1,…,ξ^N-1}⊂^N-1 such that  for every j=1,…, N-1, the function f(x'+ξ^j)-f(x') has either a maximum or a minimum in ^N-1. Let u^ be the bounded solution of (<ref>)-(<ref>). Suppose that there exists a C^2 surface  that is level surface of u^, for any > 0.
Then,  must be a hyperplane.



We apply together Theorems <ref> and <ref> .




§ SPHERICAL SYMMETRY FOR Q-MEANS-INVARIANT SURFACES
 


In this section we give applications of Theorems <ref> and   <ref>. We give characterizations of spheres, based on q-means, in the spirit of <cit.>. These results are new, even for the case p=2. 


Here, we consider those parallel surfaces _R sufficiently near , such that for every x∈_R there exists an unique y_x such that B_R(x)∩ ={y_x}, for every y_x. Also, we suppose that _1(y_x),⋯_N-1(y_x) < 1/R.



Set 1<p≤∞ and let  be a domain of class C^2 with bounded and connected boundary . 
Let u be the bounded (viscosity) solution of (<ref>)-(<ref>). 
Suppose that  is a C^2-regular surface in , that is a parallel surface to  at distance R>0.If, for some 1<q<∞ and every t>0, the function

    ∋ x↦μ_q (x,t)

is constant, then  must be a sphere.



Since  is of class C^2 and is parallel to , for every y∈, there is a unique x∈ at distance R from y.  Thus, owing to Theorem <ref>, we can infer that 

    Π_=
 
Our claim then follows from a  variant of Alexandrov's Soap Bubble Theorem (see <cit.>),  <cit.>, or <cit.>.  


The next is the elliptic counterpart to Theorem <ref>. Here, we intend that μ_q, is the q-mean of u^ on B_R(x).



Set 1<p≤∞ and let  be a domain of class C^2 with bounded and connected boundary . 
Let u^ be the bounded (viscosity) solution of (<ref>)-(<ref>). 
Suppose that  is a C^2-regular surface in , that is a parallel surface to  at distance R>0.If, for some 1<q<∞ and every > 0, the function

    ∋ x↦μ_q, (x)

is constant, then  must be a sphere.



The proof runs similarly to that of Theorem <ref>, once we replace Theorem <ref> by Theorem <ref>.









  abbrv 
  



 