Regularity for an anisotropic equation in the plane
Peter Lindqvist Diego Ricciotti
math.AP



Regularity for an anisotropic equation in the plane]
Regularity for an anisotropic equation in the plane


Peter Lindqvist,  Norwegian University of Science and Technology, Department of Mathematics, N-7491 Trondheim, Norway
peter.lindqvist@ntnu.no

Diego Ricciotti, University of Pittsburgh, Department of Mathematics, 
301 Thackeray Hall, Pittsburgh, PA 15260, USA
DIR17@pitt.edu



We present a simple proof of the C^1 regularity of p-anisotropic  functions in the plane for 2≤ p<∞. We achieve a logarithmic modulus of continuity for the derivatives. The monotonicity (in the sense of Lebesgue) of the derivatives is used. The case with two exponents is also included.

[
    Diego Ricciotti
    December 30, 2023
=====================




[2010 Mathematics Subject Classification: 35H20, 35J70.]
[Keywords: p-Orthotropic Equation, Regularity.]




§ INTRODUCTION


The minimization of the “anisotropic" variational integral

    I_Ω(v)=∫_Ω∑_i=1^n1/p_i|∂ v/∂ x_i|^p_i

over functions v(x)=v(x_1,⋯,x_n) with given values on the boundary of the bounded domain Ω⊂ℝ^n, leads to the Euler-Lagrange equation

    ∫_Ω∑_i=1^n|∂ u/∂ x_i|^p_i-2∂ u/∂ x_i∂ϕ/∂ x_i =0
 
for all test functions ϕ∈ C_0^∞(Ω). 

Denoting by =(p_1,⋯,p_n), it is required that a solution u belongs to the anisotropic Sobolev space

    W^1,(Ω) :={ u∈ W^1,1(Ω) : u_x_i∈ L^p_i(Ω) ,  i=1,⋯,n }.

Formally one has the equation

    ∑_i=1^n ∂/∂ x_i( |∂ u/∂ x_i|^p_i-2∂ u/∂ x_i)=0

in Ω.


The equation is demanding even in the plane. We shall restrict ourselves to the case n=2 and 2≤ p_1≤ p_2< ∞.
Our object is the continuity of the gradient ∇ u =(u_x_1, u_x_2) in the plane.
The recent work <cit.> of P. Bousquet and L. Brasco is devoted to the “orthotropic equation", as they call it, 

    ∂/∂ x_1( |∂ u/∂ x_1|^p-2∂ u/∂ x_1)
    +∂/∂ x_2( |∂ u/∂ x_2|^p-2∂ u/∂ x_2)
    =0

in Ω, with only one exponent 1<p<∞. They proved that u∈ C^1_loc(Ω).
Our first result is a very simple proof of the continuity of the gradient.


Let p≥ 2 and suppose that u∈ W^1,p(Ω) is a solution of (<ref>) in Ω. Then ∇ u is continuous and

    B_r(∇ u) ≤ A ( 1/R^2log(R/r)∬_B_2R|∇ u|^p )^1/p
 
where A=A(p) and B_r, B_R are concentric balls B_r⊂ B_R⊂ B_2R⊂⊂Ω.
 

The advantage of our proof is, besides its simplicity, that a modulus of continuity of the size

    (log(1/r))^-1/p
 
is provided. 
The main ingredient is an elementary inequality used by Lebesgue in 1907, valid for functions that are monotone (in the sense of Lebesgue). We exploit the fact that the partial derivative u_x_i obeys the maximum and minimum principle, a key property observed in <cit.> Lemma 2.6 .

Second, we consider the equation

    ∂/∂ x_1( |∂ u/∂ x_1|^p_1-2∂ u/∂ x_1)
    +∂/∂ x_2( |∂ u/∂ x_2|^p_2-2∂ u/∂ x_2)
    =0

in Ω, under the restriction 2≤ p_1<p_2.



Let =(p_1,p_2) with 2≤ p_1<p_2 and assume that 

u∈ W^1,(Ω). 
If u=u(x_1,x_2) is a solution of equation (<ref>), then the gradient ∇ u is continuous and

    B_r( u_x_i)
    ≤ A(1/R^2log(R/r)∬_B_2R(|∇ u|^p_1+|∇ u|^p_2))^1/p_i,

where A=A(p_1,p_2) and B_r, B_R are concentric balls B_r⊂ B_R⊂ B_2R⊂⊂Ω. The integral converges.

Here we encounter an extra difficulty. Naturally 
u_x_1∈ L^p_1(Ω), 
u_x_2∈ L^p_2(Ω),
and, consequently, u_x_2∈ L^p_1(Ω), 
but one cannot assume
u_x_1∈ L^p_2(Ω).
Indeed, the term 
|v_x_1|^p_2
is not present in the variational integral (<ref>).
This difficulty is discussed in <cit.>. Under the restriction p_2<p_1+2, this problem is settled in Proposition <ref> below, the proof of which is a direct adaptation of the method in <cit.>. By a recent result in <cit.>, the solution of (<ref>) is locally Lipschitz continuous (n=2,   2≤ p_1≤ p_2), see  Theorem 1.4 and Remark 1.5 there. By Rademacher's Theorem the gradient belongs to L^∞_loc(Ω).
Thus the integral in the right hand side is convergent also for p_2 ≥ p_1+2. Nonetheless, we have included a sketch of the proof based on the iteration in <cit.>, since the extra assumption leads to a considerable simplification. Furthermore, this approach seems to allow a generalization to the vector valued case.  
 



 §.§ Acknowledgments
 We thank Lorenzo Brasco for a discussion about <cit.>.
P.L. was partially supported by the Norwegian Research Council. He wants to thank the University of Pittsburgh for its hospitality.




§ STANDARD ESTIMATES



  
Notation.

We use standard notation.
B_r=B_r(a) denotes the ball {x∈ℝ^2  :  |x-a|<r} and when several balls like B_r, B_R appear in the same formula they are assumed to be concentric.
Usually, ∑_i means ∑_i=1^2, although the formulas in this section are valid also in n dimensions.
A variable subscript in a function denotes a derivative with respect to that variable, e.g. v_x_i=∂ v/∂ x_i and v_x_ix_j=∂^2 v/∂ x_i∂ x_j.




  
Regularization.

We shall regularize the equation so that at least second continuous derivatives are available. 
The variational integral

    I^ϵ_Ω(v)=∑_i ∬_Ω(|v_x_i|^p_i/p_i+ϵ(p_i-2) v_x_i^2/2)   ϵ >0,

has Euler-Lagrange equation

    ∑_i ∬_Ω(|u_x_i|^p_i-2u_x_i+ϵ(p_i-1)u_x_i) ϕ_x_i=0

valid for all ϕ∈ C_0^∞(Ω). Let u^ϵ∈ W^1,(Ω) denote a solution. By elliptic regularity theory, u^ϵ is smooth.






  
Estimates.

Below ξ∈ C_0^∞(Ω) is a test function, 0≤ξ≤ 1.
Recall that p_1≤ p_2.
















Let u^ϵ be a solution of (<ref>).
We have

    ∑_i ∬_Ωξ^p_2 |u^ϵ_x_i|^p_i≤ a   ∑_i ∬_Ωξ^p_2-p_i|ξ_x_i|^p_i|u^ϵ|^p_i
       +ϵ(p_2-1)p_2^2∬_Ωξ^p_2-2|∇ξ|^2|u^ϵ|^2,

where a=a(p_1,p_2).
 

Use the test function ϕ=ξ^p_2u^ϵ in (<ref>).



Let u^ϵ be a solution of (<ref>).
For ν=1, 2 we have

    ∑_i ∬_Ω(p_i-1)ξ^2|u^ϵ_x_i|^p_i-2(u^ϵ_x_ix_ν)^2   ≤ 4∑_i ∬_Ω(p_i-1)ξ_x_i^2|u_x_i^ϵ|^p_i-2(u^ϵ_x_ν)^2
         + 4ϵ(p_2-1) ∬_Ω|∇ξ|^2(u^ϵ_x_ν)^2.




We can use the derivative ϕ_x_ν in place of ϕ as a test function in (<ref>).
An integration by parts with respect to x_ν yields the differentiated equation

    ∑_i ∬_Ω(p_i-1)(|u^ϵ_x_i|^p_i-2+ϵ)u^ϵ_x_ix_νϕ_x_i
    =0.

Now use the test function

    ϕ   = ξ^2u^ϵ_x_ν
    ϕ_x_i   =ξ^2u^ϵ_x_ix_ν+2ξξ_x_iu^ϵ_x_ν

and Young's inequality.



The quantity 
|u^ϵ_x_2|^p_2-2(u^ϵ_x_1)^2
has unfavourable exponents. A bound independent of ϵ is not immediate for the term

    ∬_Ωξ_x_2^2|u^ϵ_x_2|^p_2-2(u^ϵ_x_1)^2.





Let u^ϵ be a solution of (<ref>).
We have

    ∑_i ∬_Ωξ^2 |∇( |u^ϵ_x_i|^p_i-2/2u^ϵ_x_i)  |^2≤ C ( ∑_i ∬_Ω|∇ξ|^2 |u^ϵ_x_i|^p_i-2|∇ u^ϵ|^2   
    
    +ϵ∬_Ω|∇ξ|^2|∇ u^ϵ|^2   )

where C=C(p_1,p_2).


Use

    ∂/∂ x_ν( |u^ϵ_x_i|^p_i-2/2u^ϵ_x_i) ^2
    = (p_i/2)^2|u^ϵ_x_i|^p_i-2(u^ϵ_x_ix_ν)^2

and sum over ν.






  
Convergence u^ϵ⟶ u.


Let u∈ W^1,(Ω) be a solution of equation (<ref>). 
Here we take B_R⊂⊂Ω and let u^ϵ be the solution of (<ref>)

with boundary values u on ∂ B_R.
Subtract the weak equations (<ref>) and (<ref>) and use the test function ϕ=u^ϵ-u.
After some arrangements

    ∑_i   ∬_B_R ( |u^ϵ_x_i|^p_i-2u^ϵ_x_i-|u_x_i|^p_i-2u_x_i )
    (u^ϵ_x_i-u_x_i)
       +∑_i ϵ(p_i-1)∬_B_R (u^ϵ_x_i-u_x_i)^2
       =  ∑_i ϵ(p_i-1)∬_B_R u_x_i(u^ϵ_x_i-u_x_i)
       ≤ϵ/2∑_i(p_i-1)∬_B_Ru_x_i^2
    +ϵ/2∑_i(p_i-1)∬_B_R(u^ϵ_x_i-u_x_i)^2

and the last term can be absorbed into the left-hand side.
The inequality

    2^2-p|b-a|^p≤ (|b|^p-2b-|a|^p-2a)(b-a)

yields

    ∑_i 2^2-p_i∬_B_R|u^ϵ_x_i-u_x_i|^p_i≤ϵ/2(p_2-1)∬_B_R|∇ u|^2.

The next Lemma follows from this.

Assume u∈ W^1,(Ω) solves equation (<ref>) and let u^ϵ be the solution of (<ref>) in B_R with boundary values u on ∂ B_R. Then

    u^ϵ⟶ u   uniformly in   B_R
       u^ϵ_x_i⟶ u_x_i  in   L^p_i(B_R)

as ϵ→ 0.



  It remains to establish the convergence  of the functions.
If p_1>2 it follows from Morrey's inequality in the plane that 
    u^ϵ⟶ u   uniformly in   B_R.

The case p_1=2 follows from Lemma <ref> below, since the maximum/minimum principle obviously is valid for u^ϵ. 




§ OSCILLATION OF MONOTONE FUNCTIONS

A continuous function v:Ω⟶ℝ is monotone (in the sense of Lebesgue) if

    max_Dv=max_∂ Dv  and  min_Dv=min_∂ Dv

for all subdomains D⊂⊂Ω. For the next Lemma it us enough that

    B_rv=∂ B_rv

holds for circles. Monotone functions are discussed in <cit.>.

[Lebesgue]
Let Ω⊂ℝ^2. If v∈ W^1,2_loc(Ω)∩ C(Ω) is monotone, then

    (B_r_1v)^2
    log(r_2/r_1)
    ≤π∬_B_r_2|∇ v|^2

holds for all concentric disks B_r_1⊂ B_r_2⊂⊂Ω.



As on page 388 of <cit.> an integration in polar coordinates yields

    v(r,θ_2)-v(r,θ_1)=∫_θ_1^θ_2∂ v(r,θ)/∂θ θ

for a smooth function v. 
It is enough to integrate over a half circle and use the Cauchy-Schwartz inequality to obtain

    (∂ B_r v )^2 ≤π∫_0^2π∂ v/∂θ^2 θ.

Since

    |∇ v|^2=(∂ v/∂ r)^2+1/r^2(∂ v/∂θ)^2
    ≥1/r^2(∂ v/∂θ)^2

we have

    1/r(∂ B_rv)^2
    ≤π∫_0^2π|∇ v|^2r θ

integrated over a circle of radius r.
By the monotonicity

    ∂ B_rv=B_rv≥B_r_1v

when r≥ r_1.
An integration with respect to r yields (<ref>).
The Lemma follows by approximation.


We shall apply the oscillation Lemma to the functions

    |u^ϵ_x_i|^p_i-2/2u^ϵ_x_i.

To this end, we prove that u^ϵ_x_i is monotone. This is credited to <cit.>.


Let u^ϵ denote a solution of equation (<ref>).
Then

    min_∂ B_ru^ϵ_x_ν≤ u^ϵ_x_ν(x)≤max_∂ B_ru^ϵ_x_ν

when x∈ B_r, B_r⊂⊂Ω, and ν=1, 2.


Fix ν.
Assume first that u^ϵ_x_ν≤ C on ∂ B_r, where C is a constant. We claim that u^ϵ_x_ν≤ C in B_r.

Use the test function

    ϕ^+(x)=(u^ϵ_x_ν-C)^+=max{u^ϵ_x_ν-C,  0}

defined in B_r.
Note that ϕ^+=0 on ∂ B_r and set ϕ^+=0 outside B_r.
Then ϕ^+ is admissible in the differentiated equation (<ref>). 
It follows that

    0   =∬_B_r(p_i-1)(|u^ϵ_x_i|^p_i-2+ϵ)u^ϵ_x_ix_ν(u^ϵ_x_ν-C)^+_x_i
       ≥ϵ∑_i(p_i-1)∬_B_r|(u^ϵ_x_ν-C)^+_x_i|^2

and hence

    (u^ϵ_x_ν-C)^+_x_i=0   in  B_r

for i=1, 2.
Thus (u^ϵ_x_ν-C)^+ is constant in B_r. We conclude that 
u^ϵ_x_ν≤ C in B_r as desired.

A similar proof goes for the case u^ϵ_x_ν≥ C. Now use 

    ϕ^-(x)=(C-u^ϵ_x_ν)^+.




Let u^ϵ denote a solution of equation (<ref>).
For i=1, 2 the function

    |u^ϵ_x_i|^p_i-2/2u^ϵ_x_i

is monotone in Ω.




§ THE CASE P_1=P_2≥ 2


Let p_1=p_2=p≥ 2 and let u∈ W^1,p(Ω) be a solution of (<ref>).
In order to prove Theorem <ref> we denote by u^ϵ the solution  of the regularized equation (<ref>) in B_2R⊂⊂Ω with boundary values u^ϵ=u on ∂ B_2R.
Let r≤ R. By Lebesgue's Lemma

    B_r^2( |u^ϵ_x_i|^p-2/2u^ϵ_x_i)log(R/r)
    ≤π∬_B_R|∇(|u^ϵ_x_i|^p-2/2u^ϵ_x_i)|^2.

Observe that 

    2^2-p(B_ru^ϵ_x_i)^plog(R/r)
    ≤B_r^2( |u^ϵ_x_i|^p-2/2u^ϵ_x_i)log(R/r)

by the elementary inequality (<ref>).
Choose the test function ξ in Corollary <ref> so that 0≤ξ≤ 1, ξ=1 in B_R, ξ=0 in Ω∖ B_3R/2, and |∇ξ|≤ CR^-1. Thus we can majorize the right hand side:

    ∬_B_R |∇(|u^ϵ_x_i|^p-2/2u^ϵ_x_i)|^2 ≤C_p'/R^2( ∬_B_2R|∇ u^ϵ|^p
    +ϵ∬_B_2R|∇ u^ϵ|^2)

which is uniformly bounded in ϵ (0<ϵ<1). To see this, it is enough to test equation (<ref>) with ϕ=u^ϵ-u and use Young's inequality to get

    ∑_i ∬_B_2R  |u^ϵ_x_i|^p≤ C(p)∬_B_2R|∇ u|^p+ϵ(p-1)∬_B_2R|∇ u|^2.



Since, by Lemma <ref>, u^ϵ_x_i⟶ u_x_i a.e. in B_2R as ϵ→ 0 (at least for a subsequence), we finally obtain

    (B_ru_x_i)^plog(R/r)
    ≤C_p/R^2∬_B_2R|∇ u|^p.

This concludes the proof of Theorem <ref>.


For B_4R⊂⊂Ω let  ξ∈ C_0^∞(B_4R), 0≤ξ≤ 1, ξ=1 in B_2R, and |∇ξ|≤ CR^-1.
Testing equation (<ref>) with ϕ=uξ^p_2  and using Young's inequality we obtain

    ∑_i ∬_B_4Rξ^p_2|u_x_i|^p_i≤ C(p_1,p_2) ∑_i∬_B_4Rξ^p_2-p_i|∇ξ|^p_i |u|^p_i.

Hence we can write the estimate of Theorem <ref> in the form

    (B_ru_x_i)^p ≤D_p/R^2+plog(R/r)∬_B_4R|u|^p

for r<R.




§ THE CASE 2≤ P_1<P_2<P_1+2

We shall adapt the proof in <cit.> to obtain the following summability result for the derivative of the solution u^ϵ of the regularized equation (<ref>). 
[The assumption |z|^p_1≤ f(z) on page 417, eqn. (2.2) of <cit.> is not valid here, but we have |z_1|^p_1+|z_2|^p_2≤ f(z) instead.]
We omit the details and refer to <cit.> for missing parts.


Let B_R⊂⊂Ω and let u^ϵ be a solution of equation (<ref>) in B_R.
Then there exists an exponent β=β(p_1,p_2) and a constant C=C(p_1,p_2,r,R) such that 

    ∬_B_r|u^ϵ_x_1|^p_2≤ C ( ∬_B_R(1+|u^ϵ_x_1|^p_1+|u^ϵ_x_2|^p_2))^β

for all r<R.

The proof is based on a double regularization. The Euler-Lagrange equation of the variational integral

    I^ϵ,σ_B_R(v)
    =∑_i∬_B_R(|v_x_i|^p_i/p_i+ϵ(p_i-1)|v_x_i|^2/2)
    +σ∬_B_R|v_x_1|^p_2/p_2

is

    ∬_B_R∑_i(|u_x_i|^p_i-2+ϵ(p_i-1))u_x_iϕ_x_i +σ|u_x_1|^p_2-2u_x_1ϕ_x_1=0,

for all ϕ∈ C_0^∞(B_R).
Let u^ϵ, σ denote the solution of (<ref>) with boundary values u^ϵ, σ=u^ϵ on ∂ B_R.
A similar reasoning as in <cit.>, pages 421-427,
leads for any δ, p_1≤δ<p_2, to the estimate 

    ( ∬_B_α^2R|u^ϵ, σ_x_1|^p_1/1-b)^1-b/2≤ C(p_1,p_2,δ,R,α)
    ( (∬_B_α R |u^ϵ, σ_x_1|^p_1)^1/2
    + (∬_B_α R |∇ u^ϵ, σ|^δ)^1/2)

where 0<α<1.
This is valid for every b in the range

    0<b<2-p_2/δ.

The idea is to iterate this estimate over concentric disks of radii α R, α^2R, α^3R, ...( a finite number will do) starting with δ_0=p_1 and increasing the exponent at each step.
If, for instance, 

    2κ = 2p_1-p_2/p_2-p_1p_1

we can always find an admissible b such that

    p_1/1-b=δ+κ.

Hence the powers in the iteration become p_1, p_1+κ, (p_1+κ)+κ=p_1+2κ, ..., p_1+mκ.
This yields the Lemma, but for u^ϵ,σ insted of u^ϵ.
The limit procedure σ→ 0 leads to the desired result, when one uses the minimization property

    I^ϵ, σ_B_R(u^ϵ,σ)
    ≤ I^ϵ_B_R(u^ϵ)+σ/2∬_B_R|u^ϵ_x_1|^p_2 ,

provided that we already know

    ∬_B_R|u^ϵ_x_1|^p_2 <∞.

To get rid of this restriction, we use a convenient convolution with some mollifier ρ:

    u^*=u*ρ

approximating the solution u of the original equation (<ref>).
Let u^ϵ,* be the solution of the regularized equation (<ref>) in B_R, with boundary values u^ϵ,*=u^* on ∂ B_R. 
Let u^ϵ,σ,* be a solution of (<ref>) with the same boundary values.
Then the difficult term (<ref>) can be dismissed, since now

    I^ϵ,σ_B_R(u^ϵ,σ,*)
       ≤ I^ϵ_B_R(u^*)+σ/2∬_B_R|u^*_x_1|^p_2
       ≤ I^ϵ_B_R^*(u)+σ/2∬_B_R^*|u_x_1|^p_2

where R^*>R, and R^*-R can be made as small as we please (depending on ρ in the convolution). We used the fact that the convolution is a contraction.
We now have a bound free of σ and can take the limit as σ→ 0.
The result is

    ∬_B_R|u^ϵ,*_x_1|^p_2≤ C( ∬_B_R^*(1+|u_x_1|^p_1+|u_x_2|^p_2) )^β.

As u^*=u*ρ→ u, we conclude from

    I_B_R(u^ϵ,*)≤ I^ϵ_B_R(u^ϵ,*) ≤ I^ϵ_B_R(u^*)→ I^ϵ_B_R(u)

that the weak limit in L^2(B_R) of ∇ u^ϵ,* must be ∇ u^ϵ, since the minimizer of this strictly convex variational integral is unique.
By weak lower semicontinuity

    ∬_B_r|u^ϵ_x_1|^p_2≤ C( ∬_B_R(1+|u_x_1|^p_1+|u_x_2|^p_2) )^β

since R^*→ R.
This version of inequality (<ref>) is enough for us.



§ PROOF OF THEOREM <REF>


This is almost the same as the proof of Theorem <ref>. 
For the regularized equation (<ref>) one first obtains as before, though with a few obvious changes,

    R^2(B_ru^ϵ_x_i)^p_ilog(R/r)
    ≤ A ∬_B_2R(|∇ u^ϵ|^p_1+|∇ u^ϵ|^p_2)

when r≤ R, B_2R⊂⊂Ω and for a constant A=A(p_1,p_2).

In the case p_2 < p_1 + 2 we proceed as follows. 
If B_4R⊂⊂Ω, using Proposition <ref>,  we can bound the right hand side by

    C(p_1,p_2,R) ( ∬_B_4R(1+|u_x_1|^p_1+|u_x_2|^p_2))^β(p_1,p_2)
 
which is a finite quantity independent of ϵ. The desired result follows as ϵ→ 0.

 The general case can be extracted from Theorem 1.4 in <cit.>.



The exact dependence on R is not worked out here. The result that comes from the iteration in the proof of Proposition (<ref>) above, if all steps are computed, is not illuminating.


*
abbrv



