State-constraint static Hamilton-Jacobi equations in nested domains
Yeoneung Kim Hung V. Tran Son N. T. Tu


Rate of convergence]
State-constraint static Hamilton-Jacobi equations in nested domains




The authors were supported in part by NSF grant DMS-1664424.
The second and the third authors are supported in part by NSF CAREER grant DMS-1843320.


We study state-constraint static Hamilton–Jacobi equations in a  sequence of domains {Ω_k}_k ∈ in ^n such that Ω_k ⊂Ω_k+1 for all k∈.
We obtain rates of convergence of u_k, the solution to the state-constraint problem in Ω_k, to u, the solution to the corresponding problem in Ω =  ⋃_k ∈Ω_k.
In many cases,  the rates obtained are proven to be optimal.
Various new examples and discussions are provided at the end of the paper.






[Y. Kim]

Department of Mathematics, 
University of Wisconsin Madison, 480 Lincoln  Drive, Madison, WI 53706, USA
yeonkim@math.wisc.edu


[H. V. Tran]

Department of Mathematics, 
University of Wisconsin Madison, 480 Lincoln  Drive, Madison, WI 53706, USA
hung@math.wisc.edu


[S. N.T. Tu]

Department of Mathematics, 
University of Wisconsin Madison, 480 Lincoln  Drive, Madison, WI 53706, USA
thaison@math.wisc.edu



[2010]
35B40, 
35D40, 
49J20, 
49L25, 
70H20 



[
    Son N. T. Tu
    December 30, 2023
=====================













§ INTRODUCTION

Let {Ω_k}_k ∈ be a sequence of domains in ^n such that Ω_k ⊂Ω_k+1 for all k∈.
We say that {Ω_k}_k ∈ is a sequence of nested domains.
Then, Ω = ⋃_k ∈Ω_k is also a domain in ^n.
Let H:Ω×ℝ^n→ℝ be a given continuous Hamiltonian. In this paper, we are interested in studying state-constraint solutions to the following static Hamilton-Jacobi equations:

    u_k(x) + H(x,Du_k(x)) = 0     in Ω_k, HJ_k

and

    u(x) + H(x,Du(x)) = 0     in ΩHJ.

The precise definition of state-constraint viscosity solutions is given in Section <ref>. Under some appropriate conditions, (<ref>) has a unique  state-constraint viscosity solution u_k ∈C(Ω_k) for each k ∈,
and  (<ref>) has a unique  state-constraint viscosity solution u ∈C(Ω).
Furthermore, by a priori estimates and the stability results of viscosity solutions, we have that u_k → u locally uniformly on Ω.
Our main focus here is to study how fast this convergence is in two different types of nested domains.






 §.§ Assumptions
 
In the paper, we consider the following two prototypes of nested domains, which are


  (P1) Ω_k = B(0,k) and Ω = ⋃_ k ∈ B(0,k) = ^n,

  (P2) Ω_k = B(0,1-1/k), and Ω = B(0,1). 

We list the main assumptions that will be used throughout the paper. 


  (H1) There exists C_1>0 such that

    H(x,p) ≥ -C_1      for all  (x,p)∈Ω×ℝ^n. H1



  (H2) There exists C_2>0 such that

    sup_x∈Ω |H(x,0)| ≤ C_2. H2


  (H3a) There exists a modulus ω_H:[0,∞) → [0,∞), which is a nondecreasing function satisfying ω_H(0^+) = 0 and

    |H(x,p) - H(y,p)| ≤ω_H(|x-y|(1+|p|)),
    
    |H(x,p) - H(x,q)| ≤ω_H(|p-q|),
    H3a

for x,y ∈Ω and p,q∈ℝ^n.


  (H3b) For every R>0, there exists a modulus ω_R:[0,+∞) → [0,+∞), which is nondecreasing with ω_R(0^+) = 0 and

    |H(x,p) - H(y,p)| ≤ω_R(|x-y|),
    
    |H(x,p) - H(x,q)| ≤ω_R(|p-q|),
    H3b

for x,y ∈Ω and p,q∈ℝ^n with |p|,|q|≤ R.

  (H3c) For each R>0 there exists a constant C_R such that

    |H(x,p) - H(y,p)| ≤ C_R|x-y|,
    
    |H(x,p) - H(x,q)| ≤ C_R|p-q|, 
    H3c

for x,y∈Ω and p,q ∈ℝ^n with |p|,|q|≤ R.


  (H4) H satisfies the coercivity assumption

    lim_|p|→∞(inf_x∈Ω H(x,p)) = +∞. H4


  (H5) p↦ H(x,p) is convex for each x∈Ω.

Let us give some quick comments on the assumptions here.
Assumption (<ref>) is necessary to ask a meaningful question about the rate of convergence of u_k to u.
See the discussion in Section <ref> in case where (<ref>) fails to hold.
Besides, it is clear that (<ref>) is weaker than both (<ref>) and (<ref>).





 §.§ Main results

There have been many works in the literature on the well-posedness of state-constraint Hamilton-Jacobi equations and fully nonlinear elliptic equations.
The state-constraint problem for first-order convex Hamilton-Jacobi equations using optimal control frameworks was first studied in <cit.>.
The general nonconvex, coercive first-order equations was then discussed in <cit.>.
For further developments in using optimal control formulation and obtaining optimal paths, we refer the readers to <cit.> for the finite dimensional cases, and <cit.> for the infinite dimensional cases. 
See <cit.> for discrete numerical schemes, and <cit.> for large time behavior results.
We also refer to the classical books <cit.> and the references therein.


The state-constraint problem for second-order equations was first studied in <cit.> for the Laplacian, and in <cit.> for the general possibly degenerate diffusion matrices.
Boundary behavior of blow-up solutions was discussed  in <cit.>. 
Convex solutions with state-constraint boundary were constructed in <cit.>.
The convergence of solutions to the vanishing discount problems was proved in <cit.>.




In terms of state-constraint problems in nested domains, up to our knowledge, there are only qualitative results in the literature in <cit.> where certain approximations were needed for the analysis of solutions. 
We provide here some first quantitative results on the rate of convergence of the solutions to (<ref>) as k goes to infinity in two different types ((P1) or (P2)) of nested domains.




First of all, we show that the rate of convergence is 𝒪(1/k^2) for the prototype (P1) for general nonconvex Hamiltonians.

 
Under the assumptions (P1), (<ref>), (<ref>), (<ref>), and (<ref>), we have


  (i) u(x) ≤ u_k(x) for every x∈B(0,k),

  (ii) there exists a constant C>0 depending only on H such that

    0≤ u_k(x) - u(x) ≤C(1+|x|^2)/k^2

for k ∈ and x∈B(0,k).

In particular, for any fixed R>0 and |x|≤ R,

    0≤ u_k(x) - u(x) ≤C(1+R^2)/k^2.



The condition that |x|≤ R is important since there are examples where the estimate above fails at the boundary of Ω_k. 
In Proposition <ref>, we  have, for each k ∈,  |u_k(x) - u(x)| =1 for some x∈∂Ω_k.





 
Assume (P1).
Assume further that H(x,p)=a(x)K(p) for (x,p)∈ℝ^n ×ℝ^n.
Here, K(0)=0, K is locally Lipschitz and coercive in ^n, and a ∈BUC(ℝ^n) satisfies α≤ a(·)≤β  for some given α,β>0. 
Then, u ≡ 0, and for every x∈B(0,k), we have

    0≤ u_k(x) ≤(Ce^|x|/C)e^-k/C,

where C is a constant depending only on H.
In particular, for any fixed R>0, we have

    0 ≤ u_k(x)≤(C e^R/C) e^-k/C

for every x∈B(0,R) and k≥ R. 
In addition to that, this exponential rate is optimal.




It is quite interesting to observe that we obtain the exponential rate of convergence for this particular class of nonconvex Hamiltonians and the rate is indeed optimal. When a(x) is a positive constant, the assumption K(0)=0 in the theorem above can be removed. 

 
Assume (P1).
Assume further that H(x,p)=H(p) for (x,p)∈ℝ^n ×ℝ^n.
Here, H is locally Lipschitz and coercive in ^n.
Then, u ≡ - H(0), and for every x∈B(0,k), we have

    0≤ u_k(x) - u(x) ≤(Ce^|x|/C)e^-k/C,

where C is a constant depending only on H.
In particular, for any fixed R>0, we have

    0 ≤ u_k(x) - u(x)≤(C e^R/C) e^-k/C

for every x∈B(0,R) and k≥ R. 
In addition to that, this rate is optimal.



When H(x,p) = K(p)+V(x), the analysis becomes much more complicated due to the interaction between K and V. We provide an example where the exponential rate of convergence is obtained in Example <ref>.





For convex Hamiltonians, we are able to establish the exponential rate of convergence using optimal control theory.  
Some examples for which the exponential rate is obtained are given in Proposition <ref> and Proposition <ref>.


 
Under the assumptions (P1), (<ref>),(<ref>), (<ref>), (<ref>), and (H5), we have 


  (i) u(x) ≤ u_k(x) for every x∈B(0,k),

  (ii) for each fixed x∈ B(0,k) we have

    u_k(x)  ≤ u(x)+ (C e^|x|/C) e^-k/C,

where C is a constant depending only on the growth of H.

In particular, for any fixed R>0, we have

    0 ≤ u_k(x) - u(x)≤(C e^R/C) e^-k/C

for all x∈B(0,R) and k>R.


As a byproduct, we prove the existence of a minimizer η with bounded velocity to the minimizing problem (<ref>) for each given x∈^n, which is a key element in 
 the proof of Theorem <ref>.
 Moreover, the bound on the velocity of η only depends on the growth of H and not on its smoothness.
 We believe that this bound (Theorem <ref> and Lemma <ref>)  is new in the literature.  
 See Remark <ref> for further discussions.



For the second prototype (P2), we establish the rate 𝒪(1/k) for a quite general class of Hamiltonians. 
The rate is also optimal, as pointed out in Remark <ref>.

 
Under assumptions (P2), (<ref>), (<ref>),(<ref>) and (<ref>), for any k≥ 2,

    0≤ u_k(x) - u(x) ≤C/k

for every x∈B(0,1-1/k) where C is a constant depending only on H.
Moreover, this rate is optimal.


Although we only deal with two prototype cases (P1) and (P2) in this paper, the obtained results can be extended to more general domains in a similar fashion under some appropriate conditions.
See Remarks <ref> and <ref> for example.



 §.§ Organization of the paper
 
The paper is organized in the following way. 
We first provide some results on state-constraint Hamilton-Jacobi equations needed throughout the paper in Section <ref>. 
Section <ref> and <ref> are devoted to proving Theorem <ref> and Theorem <ref>, respectively. 
In the following section, we deal with the rate of convergence for convex Hamiltonians (Theorem <ref>). 
In Section <ref>, the second prototype case is considered. 
We provide some examples and further discussion in Section <ref>. 
The proofs for some results concerning minimizers of the corresponding optimal control problem are provided in Appendix.





 §.§ Acknowledgement

We would like to thank the two referees very much for carefully reading our manuscript and giving very helpful comments to improve the presentation of the paper.




§ PRELIMINARIES
 
For an open subset Ω⊂ℝ^n, we denote the space of  bounded uniformly continuous functions defined in Ω by BUC(Ω;ℝ). 

 We say


  (i) v∈BUC(Ω;ℝ) is a viscosity subsolution of (<ref>) in Ω if for every x∈Ω and φ∈C^1(Ω) such that v-φ has a local maximum over Ω at x then v(x) + H(x,Dφ(x)) ≤ 0.


  (ii) v∈BUC(Ω;ℝ) is a viscosity supersolution of (<ref>) on Ω if for every x∈Ω and φ∈C^1(Ω) such that v-φ has a local minimum over Ω at x then v(x) + H(x,Dφ(x)) ≥ 0.


If v is a viscosity subsolution to (<ref>) in Ω, and is a viscosity supersolution to (<ref>) on Ω, that is,

    v(x) + H(x,Dv(x))    ≤ 0     in Ω,
    
    v(x) + H(x,Dv(x))    ≥ 0     on Ω

in the viscosity sense, then we say that v is a state-constraint viscosity solution of (<ref>). 



 
As pointed out in <cit.>, the state-constraint implicitly imposes a boundary condition to solutions. 
Indeed, when ∂Ω is smooth, we can define an outward normal vector ν⃗(x) at x∈∂Ω. Moreover, if the state-constraint solution v∈C^1(Ω), then v solves v(x)+H(x,Dv(x)) = 0 in Ω and satisfies

    H(x,Dv(x)) ≤ H(x,Dv(x)+βν⃗(x))     for any β≥ 0, x∈∂Ω.


If H is differentiable in p, the above condition can also be phrased as a constraint on the normal derivative  on the boundary as

    D_pH(x,Dv(x))·ν⃗(x) ≥ 0     for any  x∈∂Ω.





Now we construct a state-constraint viscosity solution to (<ref>) based on Perron's method. It is a variant of the classical result in <cit.> but we include the proof here for the sake of the readers' convenience.
Note that the Lipschitz regularity of subsolutions is encoded directly into the admissible class ℱ.

 For a real valued function w(x) define for x∈Ω, we define the super-differential and sub-differential of w at x as

    D^+w(x)   ={ p ∈^n : lim sup_y → xw(y)-w(x)-p· (y-x)/|y-x|≤ 0},
    
    D^-w(x)   ={ p ∈^n : lim inf_y → xw(y)-w(x)- p · (y-x)/|y-x|≥ 0 }.






 Assume (<ref>), (<ref>) and (<ref>). 
Then, there exists a state-constrained viscosity solution u∈C(Ω)∩W^1,∞(Ω) to (<ref>).






 Under (<ref>) and (<ref>), C_1 and -C_2 are a supersolution on Ω and a subsolution in Ω of (<ref>), respectively. By the coercivity assumption (<ref>), we can find a constant C_3>0 such that 

    H(x,p) ≤max{C_1,C_2}  for some  x∈Ω    ⟹     |p| ≤ C_3.

Let us define

    ℱ   = { w ∈C(Ω)∩W^1,∞(Ω) : -C_2≤ w(x)≤ C_1, ‖ Dw‖_L^∞(Ω)≤ C_3,
                        and w is a viscosity subsolution to  w(x)+H(x,Dw(x))≤ 0 in Ω}

and for each x∈Ω, we define

    u(x):=sup{ w(x): w∈ℱ}.

By the stability of viscosity subsolutions, we have that u is a viscosity subsolution to (<ref>) in Ω. Thus, u∈ℱ as well.

We now check that u is a viscosity supersolution to (<ref>) on Ω. Assume that  u is not a supersolution on Ω. Then, there exists x_0∈Ω, φ∈C^1(Ω) with Dφ_L^∞(B(x_0,r)≤ C_3 and r>0 such that u(x_0) = φ(x_0) and (u-φ)(x)≥ |x-x_0|^2 for all x∈ B(x_0,r)∩Ω, and 

    φ(x_0) + H(x_0,Dφ(x_0)) < 0.

From (<ref>) and (<ref>), we obtain φ(x_0)= u(x_0) < C_1. By continuity of φ and H, one can choose δ,ε∈(0,r/2) small enough so that ε < δ^2 and

    φ(x)+δ^2 < C_1,
    φ(x) + δ^2 + H(x,Dφ(x)) < 0 
        ⟹    φ(x)+ε^2 < C_1,
    φ(x) + ε^2 + H(x,Dφ(x)) < 0

for all x∈ B(x_0,2ε)∩Ω. Clearly, x↦φ(x) + ε^2 is a viscosity subsolution to (<ref>) in B(x_0,2ε)∩Ω and u(x) ≥φ(x) + ε^2 for x∈ B(x,2ε)\ B(x_0,ε). Let us define w:Ω→ℝ by

    w(x) = max{ u(x), φ(x) + ε^2}        x∈ B(x_0,ε)∩Ω,
    
    u(x)         x∈Ω\ B(x_0,ε).

Then, w(x) = max{ u(x),φ(x) + ε^2} in B(x_0,2ε)∩Ω belongs to ℱ. Therefore, w(x) is a viscosity subsolution to (<ref>). However, w(x_0) = φ(x_0)+ε^2 = u(x_0)+ε^2 > u(x_0), which is a contradiction to the definition of u.



The argument used in the proof of Perron's method implies the following corollary as well (see also <cit.>).

 Let u∈C(Ω) be a viscosity subsolution to (<ref>) in Ω. 
Assume further that v≤ u on Ω for all viscosity subsolutions v∈C(Ω) of (<ref>) in Ω. 
Then, u is a viscosity supersolution to (<ref>) on Ω.


The uniqueness of (<ref>) follows from the comparison principle. It was first studied by M. Soner in <cit.> under the following assumption on ∂Ω:
 

  (A) There exists a universal pair (r,h)∈ (0,∞)× (0,∞) and a uniformly bounded continuous function η∈BUC(Ω;ℝ^n) such that

    B(x+tη(x),rt) ⊂Ω    for all  x∈∂Ω,  t∈ (0,h]. A


See also <cit.> for other conditions to establish the comparison principle.


 
Assume (<ref>).
 If v_1∈BUC(Ω;ℝ) is a viscosity subsolution of (<ref>) in Ω, and v_2∈BUC(Ω;ℝ) is a viscosity supersolution of (<ref>) on Ω. If either 


  * (<ref>) holds, or

  * (<ref>) holds and v_2 is Lipschitz,

then v_1(x)≤ v_2(x) for all x∈Ω.

When the uniqueness of (<ref>) is guaranteed, the unique viscosity solution to (<ref>) is the maximal viscosity subsolution of (<ref>). This property will play a crucial role in dealing with the second prototype (P2).











§ A RATE OF CONVERGENCE FOR GENERAL HAMILTONIANS IN UNBOUNDED DOMAIN


In this section, we consider the first prototype (P1). 
The assumptions (<ref>), (<ref>), (<ref>) and (<ref>) are enforced throughout the section. 
By Theorems <ref> and <ref>, there exists u_k∈Lip(B(0,k)) which is the unique solution to 

    u_k(x) + H(x,Du_k(x)) ≤ 0      in B(0,k),
    
    u_k(x) + H(x,Du_k(x)) ≥ 0      on B(0,k)

in the viscosity sense. 
Based on the construction of solutions via Perron's method together with the coercivity of H, we have the following a priori estimate:

    |u_k(x)|+|Du_k(x)| ≤ C_H

for all x∈ B(0,k) in the viscosity sense. Here, C_H is a positive constant depending only on H (one can take C_H = max{C_1,C_2,C_3} from Theorem <ref>). By the Arzelà–Ascoli theorem, there is a subsequence {k_m}→∞, and a function u∈Lip(^n) such that 

    u_k_m→ u     locally uniformly in ℝ^n.




 The function u defined in (<ref>) is a viscosity solution to

    u(x) + H(x,Du(x)) = 0     in ℝ^n.

Moreover, u_k→ u locally uniformly in ℝ^n as k grows to infinity.





 
It is clear from the stability of viscosity solutions that u is a solution to (<ref>). 
The fact that u_k→ u locally uniformly in ^n follows from the uniqueness of  solutions to (<ref>).




Now we are ready to give a proof for Theorem <ref> using the doubling variables method.



 
We first note that u_k solves u_k(x)+H(x,Du_k(x)) ≥ 0 on B(0,k), and u solves u(x)+H(x,Du(x)) ≤ 0 in B(0,k) in viscosity sense. By the comparison principle, we get u_k(x) ≥ u(x) for all x∈B(0,k).

For the upper bound of u_k - u, we define the following auxiliary function

    Φ^k(x,y) = u_k(x) - u(y) - 2C_H k^2|x-y|^2 - 8C_H/k^2|y|^2

for (x,y)∈B(0,k)×ℝ^n. 
It is clear that Φ^k is bounded above by 2C_H independent of k∈ℕ. 
If |y|>k/2, then we have

    Φ^k(0,0)-Φ^k(x,y)  ≥ -u_k(x) +u_k(0) - u(0) + u(y)+2C_H k^2|x-y|^2 + 8C_H/k^2|y|^2>0,

which implies that for each k∈ℕ,  Φ^k(x,y) achieves a global maximum over B(0,k)×ℝ^n at (x_k,y_k)∈B(0,k)×B(0,k/2). 
Of course, |y_k| ≤k/2. 
Now we use Φ^k(x_k,y_k)≥Φ^k(y_k,y_k) to get

    2C_Hk^2|x_k-y_k|^2 ≤ u_k(x_k) - u_k(y_k) ≤ C_H|x_k-y_k|.

Therefore,  we deduce that

    |x_k| ≤  |y_k| + 1/2k^2 < k

for all k≥ 1 since |y_k| ≤k/2. 
Observing that x↦Φ^k(x,y_k) obtains a maximum at x_k with |x_k|<k,  we have

    u_k(x_k) + H(x_k,p_k) ≤ 0,

where p_k = 4C_H k^2(x_k-y_k) by the definition of viscosity subsolutions. 
We also observe that y↦Φ^k(x_k,y) obtains a maximum at y_k, which implies that 

    u(y) - (-2C_Hk^2|x_k-y_k|^2 - 8C_H/k^2|y|^2)

has a minimum at y_k. 
By the definition of viscosity supersolutions, we get

    u(y_k) + H(y_k,p_k+q_k) ≥ 0

where q_k = -16C_H/k^2 y_k. 
Here, it needs to be noted that

    |p_k|,|p_k+q_k| ≤ C_H,

which comes from Lipschitz continuity of u_k. 
Using (<ref>), (<ref>) and assumption (<ref>), there exists a constant C̃_H such that

    u_k(x_k) - u(y_k)    ≤ H(y_k,p_k+q_k) - H(x_k,p_k)
       = H(y_k,p_k+q_k) - H(y_k,p_k) + H(y_k,p_k) - H(x_k,p_k) 
       ≤C̃_H|q_k| + C̃_H|x_k-y_k| 
       ≤16C̃_HC_H/k^2|y_k| + C̃_H/k^2≤8 C̃_HC_H/k + C̃_H/k^2.

If we stop here, the fact that Φ^k(x_k,y_k) ≥Φ^k(x,x) for x ∈ B(0,k) gives 

    u_k(x) - u(x)≤ u_k(x_k) - u(y_k) + 8C_H/k^2|x|^2 ≤C/k + C(1+|x|^2)/k^2

for all k ≥ 2.
This gives us the rate of convergence of u_k to u is 𝒪(1k) for x∈ B(0,R), which is typically the case in light of the doubling variables method.



Nevertheless, a key new point here is to bootstrap once more to improve this rate.
The monotonicity of {u_k} allows us to bound |y_k| better.
We use that Φ^k(x_k,y_k)≥Φ^k(0,0) together with (<ref>) and u_k ≥ u to yield

    2C_H k^2|x_k-y_k|^2 + 8C_H/k^2|y_k|^2    ≤ u_k(x_k) - u_k(0) + u(0)- u(y_k) 
       ≤ u_k(x_k) - u(y_k)
       ≤16C̃_HC_H/k^2|y_k| + C̃_H/k^2.

Therefore, 

    |y_k|^2 ≤ 2C̃_H |y_k|+C̃_H/8C_H≤1/2 |y_k|^2 + 2C̃_H^2 + C̃_H/8C_H = 1/2 |y_k|^2 + C.

In particular, |y_k| ≤ C.
This bound is much better than the earlier bound that |y_k| ≤k/2.



Now for any x∈B(0,k), clearly we have that Φ^k(x_k,y_k) ≥Φ^k(x,x).
This, together with (<ref>) and |y_k| ≤ C, implies

    u_k(x) - u(x)≤ u_k(x_k) - u(y_k) + 8C_H/k^2|x|^2 ≤C(1+|x|^2)/k^2

for all k ≥ 2. 
If |x|≤ R, then

    0≤ u_k(x) - u(x) ≤C(1+R^2)/k^2,

which gives the desired result.


 
In the general setting, one only has that Φ^k(x,y) achieves a global maximum over B(0,k)×ℝ^n at (x_k,y_k) where |y_k| ≤k/2 and |x_k| < k.
In our current situation, the monotonicity of {u_k} allows us to bootstrap once more to deduce further that |y_k| ≤ C,
which helps to obtain 𝒪(1/k^2) rate of convergence.
This seems to be the best convergence rate that one is able to get through the doubling variables method here as it is unlikely that |y_k| vanishes as k →∞.

We do not know yet whether the 𝒪(1/k^2) rate of convergence is optimal or not in the general nonconvex setting.
See Questions <ref> and <ref> in Section <ref> below.









§ AN OPTIMAL RATE FOR A CLASS OF NONCONVEX HAMILTONIANS ON UNBOUNDED DOMAIN


In this section, we show that the rate of convergence u_k→ u is of order 𝒪(e^-Ck) for a class of possibly nonconvex Hamiltonians which are written as H(x,p) = a(x)K(p) with K(0)=0 and 0<α≤ a(x) ≤β. The aforementioned rate is indeed optimal. 

A brief idea for the proof is that we construct a supersolution to (<ref>) by finding a symmetric Hamiltonian H̃ such that H̃(0) = 0 and H̃≤ H. The following proposition is needed as a building block.


 Let H:ℝ^n→ℝ be defined by

    H(p) = 
    -α|p|    for  |p|≤β,
        f(p)     for  |p|≥β,

where α,β>0 and f:ℝ^n→ℝ is a coercive continuous function such that f(p) = -αβ for |p|= β and min_ℝ^n f = -αβ. Then,

    u_k(x) = αβ e^|x|-k/α

for x∈B(0,k) is the unique solution to the state-constraint problem (<ref>).


 It is clear that u_k(x) + H(Du_k(x)) = 0 in B(0,k)\{0} in classical sense. For x ∈∂ B(0,k) and φ∈C^1(B(0,k)) such that u_k-φ has a local minimum over B(0,k) at x, we have u_k(x) + H(Dφ(x)) ≥ 0 since u_k(x) = αβ = -min H. We only need to check if u_k is a viscosity supersolution at x = 0. 

Let φ∈C^1(ℝ^n) such that φ(0) = u_k(0) and u_k-φ has a local minimum over B(0,k) at x = 0. Since u_k is convex, we can replace φ by an affine function φ(x) =ξ· x+ u_k(0) for some ξ∈ℝ^n. Without loss of generality, it suffices to consider ξ≠ 0. For |x| sufficiently small, we have u_k(x) -φ(x) ≥ u_k(0) - φ(0), which implies that

    αβ e^-k/α(e^|x|/α-1) ≥ξ· x.


Now we choose x = tξ/|ξ| for t>0 small, then (<ref>) implies that αβ e^-k/α(e^t/α-1) ≥ t|ξ| for all t>0 sufficiently small. Dividing both sides by t and sending t to 0, we deduce that |Dφ(0)| = |ξ|  ≤β e^-k/α. Therefore,

    u_k(0) + H(Dφ(0)) = αβ e^-k/α -α |Dφ(0)| ≥ 0.

Consequently, u_k is the unique viscosity  solution to (<ref>).



 Since K(0) = 0, u≡ 0 is the unique solution to (<ref>). Recalling the a priori estimate ‖ u_k‖_L^∞(B(0,k))+‖ Du_k‖_L^∞(B(0,k))≤ C_H, condition (<ref>) gives

    |K(p) - K(q)| ≤ L|p-q|

for all p,q∈B(0,C_H). Let K(p_0) = min K ≤ 0 for some p_0∈ℝ^n. 
Let f(p) be a coercive, continuous function such that f(p)=-L|p_0| for |p| ≤ |p_0|, min_^n f = - L|p_0|, and f(p) ≤ K(p) for |p| ≥ |p_0|. Now we consider

    H̃(p) = 
    -  L|p|                       for  |p|≤ |p_0|,
        f(p)       for  |p|≥ |p_0|.





The graph of H̃ is described in Figure <ref>. It is clear that H̃(p)≤ K(p) for all p∈ℝ^n. Moreover, using Proposition <ref>, the unique viscosity solution to the state-constraint problem ũ_k(x) + βH̃(Dũ_k(x)) = 0 in B(0,k) is given by ũ_k(x) = β L|p_0|e^|x|-k/β L for x∈B(0,k). 


It is clear that ũ_k is also the unique viscosity solution to 1/βũ_k(x) + H̃(Dũ_k(x)) = 0 in B(0,k). Since β≥ a(x)≥α >0 and H̃≤ K, we deduce that

    1βũ_k(x) + H̃(Dũ_k(x)) ≤1a(x)ũ_k(x) + K(Dũ_k(x))

on B(0,k). Therefore, ũ_k(x)+ a(x)K(Dũ_k(x)) ≥ 0 on B(0,k). By the comparison principle, one gets

    0≤ u_k(x) ≤β L |p_0| e^|x|-k/β L

for all x∈B(0,k). The conclusion for |x|≤ R follows immediately.



In case that H(x,p)=K(p) for (x,p) ∈^n ×^n, where K is locally Lipschitz continuous and coercive in ^n, we have the unique  viscosity solution  to (<ref>) is u≡ -K(0). 
Therefore, we can assume that K(0) = 0, and Corollary <ref> follows without assuming that K(0) = 0. 



It should be noted that the local Lipschitz continuity of Hamiltonians is important when it comes to getting an exponential rate of convergence. If a Hamiltonian is only Hölder continuous around 0, we get a slower rate of convergence depending on the regularity of H as described in the following proposition. 

  Let H:ℝ^n→ℝ defined by

    H(p) = 
    -|p|^γ       if  |p|≤ 1,
     f(p)         if   |p|≥ 1,

where γ∈(0,1) and f:ℝ^n→ℝ is a continuous, coercive function with f(x) = -1 for |x|≤ 1, and min_ℝ^n f = -1. Then, the solution to (<ref>) is given by

    u_k(x) = [1-γγ(k+γ1-γ - |x|)]^γ/γ-1,      x∈B(0,k).

As a consequence, u_k→ 0 with the rate 𝒪(1/k^γ/1-γ).


 
Let us first consider the one dimensional case. 
The higher dimensional setting can be done in a same manner. 
Let μ = γ^-1, we look for a nonnegative solution to u(x)^μ = u'(x) where x∈ (0,k). We have

    u(x)^1-μ = (1- μ)x-C_k     ⟹     u(x) = (μ-1)^1/1-μ(C_k-x)^1/1-μ.

We want to choose C_k such that u'(x)∈ [0,1] for x∈ (0,k). Equivalently,

    u'(x) = u(x)^μ= (μ-1)^μ/1-μ(C_k-x)^μ/1-μ∈ [0,1]

for x∈ (0,k). Since it is an increasing function, C_k = k + 1/μ-1. Using symmetry, we guess that u_k is written as

    u_k(x) = (μ-1)^1/1-μ(k+1μ-1-|x|)^1/1-μ.

It is straightforward to see that u_k satisfies the equation in the classical sense u_k(x) -|u_k'(x)|^γ = 0 in (-k,k)\{0}. Since |u_k'(x)|≤ 1 on (-k,k)\{0}, we have u_k(x) + H(u_k'(x)) = 0 in the classical sense in (-k,k)\{0}. At |x| = k, we have u_k(x)=1 ≥ -min H. Therefore, the supersolution test at these points are satisfied. Finally, at x=0 we only need to verify the supersolution test, which is simple since if p∈ D^-u_k(0) then

    |p|≤ (μ-1)^μ/1-μ(k+μμ-1)^μ/1-μ    ⟹     u_k(0) + H(p) ≥ u_k(0) - |p|^μ≥ 0.

Thus, u_k defined above is the unique viscosity solution to the constraint problem (<ref>). Using a similar argument as in the proof of Proposition <ref>, this formula of u_k can be extended naturally to the n-dimensional case, as given in (<ref>) and the conclusion follows.


 From Proposition <ref> we see that the optimal rate of convergence can be as slow as we wish as the Hölder exponent γ→ 0^+.
This shows that the required condition  (<ref>) is really essential in this section.



When Hamiltonians are of the form H(x,p) = K(p)+V(x), the situation becomes much more complicated. See Example <ref> for a situation where we get the optimal exponential rate of convergence with nonconvex K.







§ AN OPTIMAL RATE FOR CONVEX HAMILTONIANS

In this section, the assumptions (<ref>), (<ref>), (<ref>), (H5) are always in force. The state-constraint problem was studied in the context of optimal control for convex Hamiltonians (see <cit.> for instance). When H is convex, we are able to obtain a representation formula for the viscosity solution based on the optimal control theory. Let us assume the following superlinear property (see Remark <ref> where we can remove this assumption), which is


  (H6) p↦ H(x,p) is superlinear uniformly for x∈Ω, that is,

    lim_|p|→∞(inf_x∈ΩH(x,p)/|p|) = +∞. H6


If (H5) and (<ref>) hold, then the Legendre transform L:Ω×ℝ^n of H is defined as

    L(x,v) := sup_p∈ℝ^n{ p· v - H(x,p)},     (x,v)∈Ω×ℝ^n.


 Assume (H5) and (<ref>). Then, L:Ω×ℝ^n →ℝ is continuous satisfying:


  (L1) If (H1) holds, then L(x,0) ≤ C_1 for all x∈Ω;

  (L2) If (H2 holds, then L(x,v) ≥ -C_2 for all (x,v)∈Ω×ℝ^n; 

  (L3) If (H3b) holds, then for each R>0 there exists a modulus ω̃_R(·) such that 

    |L(x,v) - L(y,v)| ≤ω̃_R(|x-y|)     for all  x,y∈Ω, |v|≤ R.


  (L5) v↦ L(x,v) is convex for each x∈Ω;

  (L6) v↦ L(x,v) is superlinear uniformly in x∈Ω, i.e.,

    lim_|p|→∞(inf_x∈ΩL(x,v)/|v|) = +∞. L6



We omit the proof of this lemma and refer the interested readers to <cit.>. 

For each x∈Ω, we define the admissible set of paths as

    𝒜_x = {η∈AC([0,∞);ℝ^n): η(0)=x and η(s)∈Ω for all s≥ 0 }

where AC([0,∞);ℝ^n) denotes the set of absolutely continuous curves from [0,∞) to ℝ^n. Note that 𝒜_x≠∅ since η(s)≡ x for all s∈ [0,∞) is an admissible path. 
From this, define the value function as

    u(x):= inf_η∈𝒜_x J[x,η]

where the cost functional is defined as

    J[x,η] = ∫_0^∞ e^-sL(η(s),-η̇(s)) ds

for (x,η)∈Ω×𝒜_x. Now we have the following classical dynamic programming principle.

 For any t>0, we have 

    u(x) = inf_η∈𝒜_x{∫_0^t e^-sL(η(s),-η̇(s)) ds + e^-tu(η(t)) }.



Using the Dynamic Programming Principle, one can prove that u∈BUC(Ω) and indeed a viscosity solution to (<ref>) as stated in the following theorems. 

 Assume (<ref>), (<ref>), (<ref>), (H5) and (<ref>), then the function u(x) defined by (<ref>) is bounded and is uniformly continuous up to the boundary, which is u∈BUC(Ω).



 The value function u∈BUC(Ω) defined in (<ref>) is a viscosity solution to the state-constraint Hamilton-Jacobi equation u(x) + H(x,Du(x)) = 0 in Ω, i.e.,

    u(x) + H(x,Du(x)) ≤ 0     in Ω,
    
    u(x) + H(x,Du(x)) ≥ 0     on Ω.



We omit the proofs of Theorems <ref>, <ref> and <ref>. 
We refer to  <cit.> for those who are interested. 


On the other hand, when Ω = ℝ^n, it is known that the function u(x) defined in (<ref>) satisfies the Hamilton-Jacobi equation (<ref>) in viscosity sense (see <cit.> for instance). 

 For each x∈ℝ^n, we define

    u(x) = inf_η∈𝒜_x∫_0^∞ e^-sL(η(s),-η̇(s)) ds

subject to 𝒜_x = {η∈AC([0,∞);ℝ^n): η(0) = x  }. Then, u∈BUC(ℝ^n) is a viscosity solution to (<ref>) and we have the following priori estimate:

    ‖ u‖_L^∞(ℝ^n) + | Du ‖_L^∞(ℝ^n)≤ C_H.





 
We may assume that H is just coercive rather than superlinear. 
When a Hamiltonian is coercive, we still have that (<ref>) holds for some C=C_H>0. 
Therefore, for |p| ≥ C, we can modify H so that (<ref>) holds. Furthermore, we can impose a quadratic growth rate on H as following.


  (H7) There exist some positive constants A,B such that

    A^-1|v|^2 - B ≤ H(x,p) ≤ A|v|^2 + B     for  (x,p)∈ℝ^n×ℝ^n. H7


It is easy to see from (H7) that we have (4A)^-1|v|^2 - B ≤ L(x,v) ≤ 4A|v|^2+B for all (x,v)∈ℝ^n×ℝ^n. By making A bigger, we can assume the following.


  (L7) 

    A^-1|v|^2 - B ≤ L(x,v) ≤ A|v|^2 + B     for (x,v)∈ℝ^n×ℝ^n. L7









We give a proof for the existence of a minimizer with bounded velocity to (<ref>) for the sake of readers' convenience in Appendix. 
This is an extremely important fact in our analysis and is a key element in the proof of Theorem <ref> (see Remark <ref> for further discussions).
To establish this point, the following lemma on the subdifferentials of L(x,v) in v is needed. For continuously differentiable Lagrangians, it is obvious, but we state here a slightly more general version.


 Let L:ℝ^n×ℝ^n →ℝ be continuous and satisfy (L5) and (<ref>). There exists C_L >0 such that for all v∈ℝ^n, we have

    |ξ|≤ C_L(1+|v|)     whenever ξ∈ D^-_v L(x,v).




For simplicity, let us assume further that 


  (L8) (x,v)↦ L(x,v) is continuously differentiable on ℝ^n×ℝ^n.

This assumption can be removed in the proof of Theorem <ref> due to the fact that the estimate (<ref>) does not depend on the regularity of H, hence, we can approximate H by convex, smooth Hamiltonians.


 Let L(x,v) be a continuous Lagrangian satisfying (L5), (<ref>) and (L8). Then, for each x∈ℝ^n, there exists η∈𝒜_x such that J[x,η] = u(x) and also ‖ e^-s/2η̇(s)‖_L^2(0,∞)≤ C_4 where C_4 depends only on C_H,A,B.


The existence of minimizers of smooth Lagrangian is sufficient for our proof of Theorem <ref> since the last estimate does not depend on the smoothness of L or H. Clearly, a minimizer for a general continuous Lagrangian can be obtained via approximation of smooth Lagrangians (see Appendix).


A minimizer to (<ref>) satisfies the following properties.



 Let x∈ℝ^n and η be a corresponding minimizer. For any t>0, we have

    u(x) = ∫_0^t  e^-s L(η(s),-η̇(s)) ds  + e^-tu(η(t)).

Furthermore, for every t,h>0, we have 

    u(η(t)) = e^t∫_t^∞ e^-sL(η(s),-η̇(s)) ds

and

    e^-tu(η(t)) = ∫_t^t+h  e^-s L(η(s),-η̇(s)) ds  + e^-(t+h)u(η(t+h)).





 Let x∈ℝ^n and η be a minimizer to (<ref>) associated with it. Then, there exists a constant C_5>0 depending only on C_H,A,B such that |η̇(s)|≤ C_5 for a.e. s∈ (0,∞).





 We provide here a connection between a minimizer η of u(x) = J[x,η] and some properties in the view of the method of characteristics. If H is assumed to be C^2, then L∈C^2 and η is a weak solution to the Euler-Lagrange equation 

    D_xL(η(s),-η̇(s)) - D_vL(η(s),-η̇(s)) + d/ds(D_vL(η(s),-η̇(s))) = 0.

Assume that η∈C^2 (it holds if, for instance L∈C^2,α for some α∈ (0,1)). Then, one can define the momentum p(s) = D_vL(η(s),-η̇(s)) and show that

    u(η(t)) + H(η(t),p(t)) = 0

for t>0. Indeed, for every fixed x∈ℝ^n, we recall that

    v ∈ D_p^-H(x,p)   ⇔   p ∈  D^-_vL(x,v)   ⇔   H(x,p) + L(x,v) = p· v.

Using (<ref>) we can deduce that 

    d/ds(e^-sp(s))    = e^-sD_xL(η(s),η̇(s)),
        d/ds(H(η(s),p(s)))    = -η̇(s)·p(s),
    d/ds(e^-sH(η(s),p(s)))    = e^-sL(η(s),-η̇(s)).

From that, we can derive the characteristic ODEs for s>0, which are

    -η̇(s)    = D_pH(η(s),p(s)),
       ṗ(s)    = p(s) - D_xL(η(s),-η̇(s)).

This together with (<ref>) yields that

    u(η(t)) + H(η(t),p(t)) = Ce^t     where     C = lim_a→∞ e^-aH(η(a),p(a)).

Lemma <ref> together with Lemma <ref> gives us a uniform bound on p, thus C = 0. Hence, (<ref>) follows.






Now we give a proof for Theorem <ref>. Recall that we have the value function

    u_k(x) =  inf_η∈𝒜_x^k∫_0^∞ e^-sL(η(s),-η̇(s)) ds,

where 𝒜^k_x = {η∈AC([0,∞);ℝ^n): η(0) = x and η(s)∈B(0,k) for s≥ 0 }. Then, u_k solves the state-constraint problem (<ref>).


 Let k ∈ be given. We may assume that H satisfies (<ref>) and (<ref>) up to modification for |p| large enough. Also, since the final estimate does not depend on the smoothness of L, we can assume H is smooth and thus L is smooth without any loss of generality. Clearly, 𝒜_x^k ⊂𝒜_x for any x∈B(0,k), which implies that u_k(x) ≥ u(x). 


For x∈B(0,k), let η∈𝒜_x be a minimizer to (<ref>), if η(s)∈B(0,k) for all s>0, then η∈𝒜_x^k as well, hence u(x) = u_k(x). Otherwise, there exists t>0 such that η(t)∈∂ B(0,k) and η(s)∈ B(0,k) for all s∈ (0,t). By Lemma <ref>, we have

    k =|η(t)| ≤ |η(0)| + ∫_0^t |η̇(s)| ds ≤ |x| + C_5t,

which implies that t≥k-|x|/C_5. Let us define

    γ(s) = η(s)    if  s∈ [0,t],
    η(t)    if  s∈ [t,∞),

so that γ∈𝒜_x^k.
Using Lemma <ref>, we have

    u(x)    = ∫_0^t e^-sL(η(s),-η̇(s)) ds + e^-tu(η(t)) 
       ≥∫_0^t e^-sL(γ(s),-γ̇(s)) ds - C_H e^-t
       ≥∫_0^∞ e^-sL(γ(s),-γ̇(s)) ds - C_1 e^-t- C_He^-t
       ≥ u_k(x) - ((C_1+C_H) e^|x|/C_5) e^-k/C_5.

Consequently, we obtain (<ref>). The conclusion for |x|≤ R follows immediately.



Here, we note that the constants in the proof above do not depend on the regularity of the Lagrangian. As long as a minimizer exists, we get the same exponential rate of convergence. See Appendix for a discussion on the existence of minimizers.
It is worth noting here that, for each x∈B(0,k), the existence of a minimizer  η∈𝒜_x  to  (<ref>) with bounded velocity is a nontrivial fact and plays an essential role in the proof above.
Moreover, the bound on the velocity of η only depends on C_H, A, B.


In the rest of this section, we provide two explicit examples to show that the rate 𝒪(e^-k/C) is indeed optimal.



 §.§ Examples with exponential rate of convergence


 Let H(p):ℝ→ℝ be defined by H(p) = |p-1|-1 for p∈ [0,2] and H(p)≥ 0 elsewhere such that H is continuous and coercive. Let u_k be the solution to (<ref>) on [-k,k], then u_k→ 0 locally uniformly on ℝ as k→∞. Here, u≡ 0 is the unique solution to (<ref>). Furthermore, we have u_k(k)=1 for all k∈ℕ and

    u_k(x) ≥ e^-2k on [-k,k].



 It is clear that u_k(x) = e^x-k solves v(x)+H(v'(x)) = 0 in (-k,k) in the classical sense, and indeed, in viscosity sense. We need to verify that u_k is a viscosity supersolution on [-k,k]. Let u_k-φ has a local minimum at x=-k for φ(x)∈C^1 (). Clearly, we can see that 

    φ'(-k) ≤ u_k'(-k) = e^-2k,

which implies
e^-2k + H(φ'(-k)) ≥ 0. On the other hand, at x=k, one has

    u_k(k)+H(φ'(k)) =1+H(φ'(k)) ≥ 0

since by definition of H, it is bounded below by -1. Therefore, u_k(x) = e^x-k is the unique viscosity solution to (<ref>), and furthermore e^-2k≤ u_k(x) ≤(e^|x|)e^-k for all x∈ [-k,k]. In addition to that, we have u_k(k) = 1 for all k ∈, hence, the convergence fails when x = k.




 §.§ Optimal control formulations



We give another example  from the optimal control theory point of view (see <cit.>). 
Let us recall briefly the setting of optimal control as follows. Let U be a compact metric space. 
We regard a control as a Borel measurable map α:[0,∞)↦ U. 
Let Ω be an open subset of ℝ^n with the connected boundary satisfying (<ref>).
 We also assume that b = b(x,a): Ω× U→ℝ^n, f = f(x,a):Ω× U→ℝ satisfy 

    sup_a∈ U|b(x,a) - b(y,a)| ≤ L(b)|x-y|       for all x,y∈Ω,
       sup_a∈ U|b(x,a)|≤ K(b)      for all x∈Ω,
       sup_a∈ U|f(x,a) - f(y,a)| ≤ω_f(|x-y|)      for all x,y ∈Ω, 
       sup_a∈ U|f(x,a)| ≤ K(f)      for all x ∈Ω,

where K(b), L(b),K(f) are positive constants and ω_f is a nondecreasing continuous function with ω_f(0^+) = 0.


For each x∈Ω and a given control  α(·):[0,∞)→ U, let y^x,α(t) be a controlled process (we will write α instead of α(·) as a control for simplicity), which is a solution to

    d/dt y^x,α(t) = b(y^x,α(t),α(t))      for  t >0,
        y^x,α(0) = x.

We denote the set of controls (strategies)  where y^x,α(t)∈Ω for all t≥ 0 and y^x,α solves the ODE above by 𝒜_x. The value function is defined by

    u(x) = inf_α∈𝒜_x∫_0^∞ e^-t f(y^x,α(t),α(t)) dt.

Here, one can define the Hamiltonian associated with b and f as

    H(x,p) := sup_a∈ U{ -b(x,a)· p - f(x,a)}∈C(Ω×ℝ^n;ℝ).

It was proved in <cit.> that u is a viscosity solution to (<ref>).

 Let n=1 and U = [-1,1]. Let us consider the following Hamiltonian defined as

    H(x,p) = sup_a ∈ [-1,1]{ -a· p -e^-|x|} = |p| - e^-|x|,      (x,p)∈ℝ×ℝ.

Then, the solution to (<ref>) is given by u_k(x) = e^-|x|/2 + e^|x|-2k/2 for x∈ [-k,k], while the solution to (<ref>) is u(x) = e^-|x|/2. Hence, the exponential rate of convergence is obtained.


 In the optimal control setting, the Hamiltonian above is obtained by considering U=[-1,1], b(x,a) = a and f(x,a) = e^-|x|. To find u_k(x_0) and u(x_0), one needs to find a control α(t) that minimizes

    ∫_0^∞ e^-s-|y(s)| ds    subject to    ẏ(t)    = α(t) ∈ [-1,1],
    
    y(0)    = x_0.

It is easy to see the following points:


  (i) An optimal control for the unconstrained problem with x_0≥ 0 (x_0<0)  is α(t) ≡ 1 (α(t) ≡ -1, respectively).

  (ii) An optimal control for the constrained problem on [-k,k] with x_0≥ 0 (x_0<0) is α(t) ≡ 1 on [0,k-x_0] and 0 elsewhere (α(t) ≡ -1 on [0,k+x_0] and 0 elsewhere, respectively).

Once we have the optimal controls, we can easily compute the value function and the result follows. In conclusion, for all x∈ [-k,k] we have

    0≤ u_k(x)-u(x)= (e^|x|/2)e^-2k.

In this example, the convergence holds everywhere in [-k,k] with the rate 𝒪(e^-k).



 
One interesting fact to point out here is that the optimal path starting from x_0 for the state-constraint problem on [-k,k] in Proposition <ref>
stays on the boundary ± k for all t ≥  k - |x_0|.





§ THE CASE OF BOUNDED DOMAIN

The second prototype case is considered in this section. 
Let us assume that (P2), (<ref>), (<ref>), (<ref>) and (<ref>) are enforced. 
Recall that Ω_k = B(0,1-1/k) for k ∈, and Ω = B(0,1). 
Let u_k∈Lip(Ω_k) be the unique viscosity solution to

    u_k(x) + H(x,Du_k(x))    ≤ 0     in Ω_k,
    
    u_k(x) + H(x,Du_k(x))    ≥ 0     on Ω_k.

It is clear that we still have the following priori estimate

    ‖ u_k‖_L^∞(Ω_k) + ‖ Du_k‖_L^∞(Ω_k)≤ C_H.



 For each k∈ℕ, let u_k be the unique solution to (<ref>). Then, there exists u ∈ BUC (Ω) such that u_k→ u locally  uniformly on Ω as k grows to infinity. Moreover, u has the same bounds as in (<ref>) and solves

    u(x) + H(x,Du(x))    ≤ 0     in Ω,
    
    u(x) + H(x,Du(x))    ≥ 0     on Ω

in viscosity sense.


 From a priori estimate (<ref>), by Arzelà-Ascoli's theorem and a diagonal argument we can extract a subsequence such that u_k_m→ u uniformly on compact subsets of Ω. By the stability of viscosity solutions we obtain that u∈C(Ω) is a viscosity solution to

    u(x) + H(x,Du(x)) = 0    in Ω.

We deduce that |u(x)|≤ C_B and |u(x)-u(y)| ≤ C_H|x-y| for x,y∈Ω. We can extend u∈Lip(Ω) with the same priori bound as in (<ref>). We need to show that u is a viscosity supersolution to u(x) + H(x,Du(x)) =  0 on Ω. 

We can verify it using Corollary <ref>. Indeed, let v∈C(Ω) be a viscosity subsolution to (<ref>) in Ω. Applying the comparison principle to u_k(x) + H(x,Du_k(x)) ≥ 0 on Ω_k, we have that v(x) ≤ u_k(x) for x∈Ω_k. Now fixing r∈ (0,1), we have v(x) ≤ u_k(x) for all x∈B(0,r) and r ≤ 1-1/k if k is large enough. Letting k→∞, we deduce that v(x)≤ u(x) for x∈B(0,r). Since we have u,v∈C(B(0,1)), the inequality v≤ u on B(0,1) follows. Hence, u is a viscosity supersolution to (<ref>) by Corollary <ref>.


Now we are ready to give a proof for Theorem <ref>. 
We note that star-shaped and scaling properties of {Ω_k} play an important role. 


 The fact that u_k(x)≥ u(x) on Ω_k is clear by the comparison principle. For k≥ 2, let us define 

    ũ_k(x):= k/k-1u_k(k-1/kx)    for x∈B(0,1).

It is clear that ũ_k is a viscosity subsolution to 

    k-1/kũ_k(x) + H(k-1/kx, Dũ_k(x)) = 0     in   B(0,1).

From (<ref>) and (<ref>), there exists C̃_H such that |H(x,p) - H(x,p)|≤C̃_H|x-y| for all x,y∈Ω and |p| ≤ C_H. Therefore, by using (<ref>) we have

    ũ_k(x) + H(x,Dũ_k(x))    ≤1/kũ_k(x) +H(x,Dũ_k(x)) -  H(k-1/kx,Dũ_k(x))≤C_H+C̃_H/k

for all x∈ B(0,1). By the comparison principle and the fact that u solves (<ref>) in the viscosity sense, we deduce that

    ũ_k(x)-C_H+C̃_H/k≤ u(x)     for all  x∈B(0,1).

Consequently, we obtain the conclusion u_k(x) ≤ u(x) + C/k for x∈Ω_k where the constant C can be chosen as C = 2C_H+C̃_H.










   

  (i) It is clear from the proof that prototype condition (P2) can be relaxed as following. 

  (P2') Assume 0 ∈Ω_k for all k ∈, Ω = ⋃_k ∈Ω_k is bounded, and the comparison principle for the state-constraint problem holds on Ω_k,Ω.
Assume further that, for k ∈,

    (1-1/k)Ω⊂Ω_k.



  (ii) Theorem <ref> can also be proved using the doubling variable method with the following auxiliary function (see <cit.>)

    Φ^k(x,y): = k+1k-1 u_k(k-1k+1x) - u(y) - C_H k^2|x-y|^2

for (x,y)∈(1+1k)Ω×Ω.



The following remark shows that the rate 𝒪(1/k) is indeed optimal.

 
Let H be defined as in Proposition <ref>, we see that u_k(x) = e^x-(1-1/k) solves (<ref>) and u(x) = e^x-1 solves (<ref>), therefore

    0≤ u_k(x) -u(x) =e^x-1(e^1/k-1)≤2/k

for x∈[-(1-1/k),1-1/k]. Besides, e^1/k - 1 ≥1/k, and so, 𝒪(1/k) is optimal.





§ DISCUSSIONS

We give here some further discussions along the line with the topics considered in the paper.
Firstly, when our Hamiltonian is given as H(x,p)=a(x)K(p) in the first prototype (P1), we get an exponential rate of convergence provided that the assumption (<ref>) is enforced (Theorem <ref>). 
Without this assumption, we have an example with a polynomial rate of convergence whose power can be increased or decreased as much as we want. 

 Let us consider n=1, H(x,p) = (1+|x|/m)K(p) for m>1 and K:ℝ→ℝ defined by 

    K(p)= 
    -|p|    for  |p|≤ 1,
    
    |p| - 2    for  |p|≥ 1.

The unique viscosity solution to (<ref>) is

    u_k(x) = (1+|x|)^m/m(1+k)^m-1    for  x∈ [-k,k].

Clearly, u_k(x)→ 0 locally uniformly with rate 𝒪(1/k^m-1) for any given m>1. 
We should note that the limit 0 is not a unique solution to (<ref>). 
Another solution to  (<ref>) is u(x) =m^-1(1+|x|)^m, but it does not belong to ().




 
Assume n=1, H(x,p)=K(p) + V(x) where V(x) = e^-|x| and K:ℝ→ℝ defined by

    K(p)= 
    -|p|      for  |p|≤ 1,
    
    |p|-2     for  |p|≥ 1.

The unique state-constraint viscosity solution to (<ref>) is

    u_k(x) = -1/2e^-|x| + (e^-k-1/2e^-2k)e^|x|,      x∈ [-k,k],

and the unique viscosity solution to (<ref>) is

    u(x) = -1/2e^-|x|,      x∈ℝ.

We have u_k→ u locally uniformly in ℝ with rate 𝒪(e^-k).


















Secondly, prototype condition (P1) can be relaxed as follows.
   It is clear from the proofs of our main results (Theorems <ref>, <ref>, <ref>, and  Corollary <ref>) that prototype condition (P1) can be relaxed as following. 

  (P1') Assume Ω_k is bounded, B(0,k) ⊂Ω_k, and the comparison principle for the state-constraint problem holds on Ω_k for all k ∈.
Of course, Ω = ⋃_k ∈Ω_k = ^n here.





Thirdly, there are some open questions we are not able to answer yet. 


 
In the first prototype (P1) case, what is the optimal rate of convergence of u_k to u in the general nonconvex setting?


A more specific question is as follows.


Assume (P1), and H(x,p) = K(p)+V(x), where K ∈(^n) is coercive and nonconvex, and V∈BUC(ℝ^n).
Is it true that we always have an exponential rate of convergence of u_k to u?






§ APPENDIX
  




 §.§ Proofs of some lemmas



 We first prove the result for all (x,v) with |v|≤ 1, then by scaling we get the result for all (x,v). Using (<ref>) we have -B ≤ L(x,v) ≤ 4A+B for all (x,v) with |v|≤ 2. For u,v∈ B(0,1) with u≠ v, let w = v + |v-u|^-1(v-u). Then, |w|< |v| + 1 < 2. Let λ = (1+|u-v|)^-1∈ (0,1), we have v = λ u + (1-λ)w. By the convexity, one obtains

    L(x,v) - L(x,u)    ≤ (1-λ)(L(x,w) - L(x,u)) ≤ (4A+2B)|u-v|.

By symmetry, we deduce that |L(x,u) - L(x,v)| ≤ (4A+2B)|u-v| for all (x,v) with |v|≤ 1. In other words, we have that |ξ| ≤ 4A+2B whenever ξ∈ D^-_v L(x,v) for (x,v)∈ℝ^n×B(0,1). Now for r>1, we define L_r(x,v) = r^-2L(x,rv) for (x,v)∈ℝ^n ×ℝ^n. We observe that

    A^-1|v|^2 - B≤ A^-1|v|^2-B r^-2≤   L_r(x,v)
    ≤ A|v|^2 + B r^-2≤ A|v|^2+B

for all (x,v). For v∈ℝ^n with |v|≥ 1, let r=2|v|>1 and u = v/2|v|∈ B(0,1) so that v = ru. Since ξ∈ D_v^-L(x,v) implies ξ/r∈ D_u^-L_r(x,u), we have |ξ| ≤ (4A+2B)|r| = (8A+4B)|v|.













 Let {η_k}_k=1^∞⊂𝒜_x be a minimizing sequence in AC([0,∞)) such that lim_k→∞ J[x,η_k] = u(x). From the uniform boundedness of u and the quadratic bounds of L(x,v), we have

    ‖ e^-s/2η̇_k(s) ‖_L^2((0,∞);ℝ^n)≤ C_4.

Here, C_4 can be chosen as (A(2C_H+B))^1/2. By the weak compactness of L^2, there exists g such that e^-s/2g(s) ∈ L^2((0,∞);ℝ^n) and a subsequence {k_j} →∞ such that e^-s/2η̇_k_j⇀ e^-s/2g weakly in L^2((0,∞);ℝ^n) as j→∞. 


Writing g as e^s/2g · e^-s/2 and using the Cauchy-Schwartz inequality, we get g∈ L^1_loc((0,∞);ℝ^n). For t>0, we let η(t) = x+∫_0^t g(s) ds. Clearly, η∈𝒜_x and one obtains that η_k_j→η pointwise with η̇=g almost everywhere. On the other hand, the convexity of L implies

    L(η_k_j(s),-η̇_k_j(s)) ≥ L(η_k_j(s),-η̇(s)) - D_vL(η_k_j(s),-η̇(s))·(η̇_k_j(s) - η̇(s)).

Therefore, 

    ∫_0^∞ e^-s L(η_k_j(s),-η̇_k_j(s))ds    ≥∫_0^∞ e^-s L(η_k_j(s),-η̇(s))ds 
             + ∫_0^∞ e^-s/2D_vL(η_k_j(s),-η̇(s))· e^-s/2(η̇_k_j(s) - η̇(s))ds.

Since |D_vL(η_k_j(s),-η̇(s))| ≤ C_L(1+|η̇(s)|) for a.e. s∈ (0,∞), it is clear that 

    e^-s/2D_vL(η_k_j(s),-η̇(s)) → e^-s/2D_vL(η(s),-η̇(s))

in L^2((0,∞);ℝ^n) and thus

    ∫_0^∞ e^-sD_vL(η_k_j(s),-η̇(s))·(η̇_k_j(s) - η̇(s))ds

converges to 0 as k goes to infinity, which yields that J[x,η] ≤ u(x). Hence J[x,η] = u(x).





 By the definition of u in (<ref>), we have 

    u(η(t)) ≤∫_0^∞  e^-s L(γ(s),-γ̇(s)) ds = e^t ∫_t^∞ e^-ξL(η(ξ),-η̇(ξ)) dξ,

where γ(s) = η(t+s) for s≥ 0. Thus,

    e^-tu(η(t)) ≤∫_t^∞ e^-ξL(η(ξ),-η̇(ξ)) dξ.

By the dynamic programming principle and (<ref>), we have

    u(η(0))    ≤∫_0^t  e^-sL (η(s),-η̇(s)) ds  + e^-tu(η(t)) ≤∫_0^∞  e^-s L(η(s),-η̇(s)) ds =  u(η(0)).

Therefore, (<ref>), (<ref>) and (<ref>) follow.






 For every t,h>0, by Lemma <ref> we have that

    e^-tu(η(t)) - e^-(t+h)u(η(t+h))/h  = 1/h∫_t^t+h  e^-s L(η(s),-η̇(s)) ds .

Let φ∈C^1(ℝ) such that u-φ has a local min at η(t) and u(η(t)) = φ(η(t)), then

    e^-tu(η(t)) - e^-(t+h)u(η(t+h))/h≤e^-tφ(η(t)) - e^-(t+h)φ(η(t+h))/h.

Therefore,

    1/h∫_t^t+h  e^-s L(η(s),-η̇(s)) ds  ≤e^-tφ(η(t)) - e^-(t+h)φ(η(t+h))/h.

Since η(t) is differentiable a.e. in (0,∞), at those t where η(t) is differentiable, let h→ 0^+ we deduce that

    e^-tL(η(t),-η̇(t)) ≤ -d/dt(e^-tφ(η(t))) = e^-tφ(η(t)) - e^-tDφ(η(t))·η̇(t).

Thus, for a.e. t>0 where η is differentiable, we have

    L(η(t),-η̇(t)) ≤φ(η(t)) - Dφ(η(t))·η̇(t).

By (<ref>) and  a priori estimate (<ref>) for a.e. t∈ (0,∞) we have that 

    A^-1|η̇(t)|^2 - B ≤φ(η(t)) - Dφ(η(t))·η̇(t) ≤ C_H+C_H|η̇(t)|.

This shows that |η̇(t)|≤ C_5 for a.e. t∈ (0,∞), and C_5 only depends on C_H,A,B. 


It is worth emphasizing again here that the bound C_5 on the velocity of η only depends on C_H, A, B, which can be seen clearly from the last chain of inequalities in the above proof.
In fact, one can choose explicitly that C_5 = (2AB+2AC_H+A^2C_H^2)^1/2.





 §.§ Existence of minimizers in the general case

We show that one can remove the smoothness of L in Theorem <ref> under the assumption (L3).


Let us consider mollifiers in ℝ^2n defined as {η_ε}_ε>0 such that η_ε(x) = 1/ε^2nη(x/ε) for x∈ℝ^2n where η∈C_c^∞(ℝ^2n) satisfying 0≤η≤ 1, supp (η) ⊂ B_ℝ^2n(0,1) and ∫_ℝ^2nη(x) dx = 1.

For each ε>0 we define the convolution L^ε = η_ε*L ∈C^∞(ℝ^n×ℝ^n). It is easy to see that L^ε is bounded below, (L5), (L6) are preserved to L^ε and (L7) now becomes:


  (L7^ε) There exist positive constants 
A_ε, B_ε such that A^-1_ε - B^-1_ε≤ L^ε(x,v) ≤ A_ε|v|^2 + B_ε for all (x,v)∈ℝ^n×ℝ^n, and A_ε→ A, B_ε→ B as ε→ 0.

By Theorem <ref>, there exists a minimizer γ_ε in 𝒜_x such that 

    u^ε(x):= inf_ζ∈𝒜_x∫_0^∞ e^-sL^ε(ζ(s),-ζ̇(s)) ds  = ∫_0^∞ e^-sL^ε(γ_ε(s),-γ̇_ε(s)) ds.

Let H^ε be the Legendre transform of L^ε.
Then, we can show that u^ε is the unique solution to u^ε(x)+H^ε(x,Du^ε(x)) = 0 in ℝ^n.  It is easy to see that H^ε→ H locally uniformly in ℝ^n×ℝ^n, therefore by stability of viscosity solutions, u^ε→ u locally uniformly in ℝ^n as ε→ 0.


We indeed have that γ^ε is smooth according to Remark <ref>. Furthermore, Theorem <ref> yields that ‖ e^-s/2γ̇_ε(s)‖_L^2≤ C and |γ̇_ε| ≤ C pointwise in (0,∞). Therefore, we can define γ∈𝒜_x such that (up to subsequence) γ_ε→γ locally uniformly on [0,∞) and e^-s/2γ̇_ε⇀ e^-s/2γ̇ weakly in L^2. Since L^ε→ L uniformly on a compact set and { -γ̇_ε(s)} _ε>0 is bounded, we obtain that

    L^ε(γ_ε(s),-γ̇_ε(s)) = L(γ(s),-γ̇_ε(s)) + ω̃_C_5(ε)+ω̃_C_5(|γ_ε(s)-γ(s)|)

using (L3). Therefore, it suffices to show that

    ∫_0^∞ e^-s L(γ(s),-γ̇(s)) ds ≤lim inf_ε→ 0∫_0^∞ e^-sL(γ(s),-γ̇_ε(s)) ds.

For simplicity, let dμ = e^-sds be a probability measure on [0,∞). It is easy to see that the functional I:L^2(μ)→ℝ maps f↦∫_0^∞ L(γ(s),f(s))dμ(s) is convex and lower semicontinuous, thus it is also weakly lower semicontinuous. Now since γ̇_ε⇀γ̇ weakly in L^2(dμ), we obtain (<ref>) and thus γ is a minimizer for u(x).

 Inequality (<ref>) for the  Cauchy problem (finite time horizon) is proved using a different argument by H. Ishii in <cit.> under more general assumptions. Such inequalities are crucial for the analysis of large time behavior of solutions to the time-dependent problems.















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