Existence of Viscosity Solutions to a Parabolic Inhomogeneous Equation Associated with Infinity Laplacian

In this paper, we obtain the existence result of viscosity solutions to the initial and boundary value problem for a nonlinear degenerate parabolic inhomogeneous equation of the form , where denotes infinity Laplacian given by .

• Infinity Laplacian
• Viscosity Solution
• Inhomogeneous Equation

1. Introduction

In this paper, we consider the nonlinear degenerate parabolic inhomogeneous equation involving infinity La- place

where

denotes the 3-homogeneous infinity Laplacian. We want to establish the existence result of viscosity solutions to the initial and Dirichlet boundary problem.

The homogeneous infinity Laplace equation is the Euler-Lagrange equation associated with - variational problem. See for details [1] -[5] and the references therein. Recently, Juutinen and Kawohl [6] con- sidered the degenerate and singular parabolic equation

They proved the existence and uniqueness for both Dirichlet and Cauchy problems, established interior and boundary Lipschitz estimates and a Harnack inequality, and also provided numerous explicit solutions. Due to the degeneracy and the singularity of the Equation (2), they introduced the approximating equations to obtain the existence result with the aid of the uniform continuity estimates. And in [7] we considered the corresponding inhomogeneous parabolic equation. Notice that the 1-homogeneous infinity Laplacian

is related to game theory named tug-of-war [8] . In [9] -[12] , Akagi, Suzuki, et al. considered the following degenerate parabolic equation

.

They also introduced the corresponding approximating equations and got the uniform continuity estimates of approximate solutions by the barrier function arguments. By this approximate procedure, the existence of the solutions was obtained. They also proved the uniqueness and the asymptotic behavior of the viscosity solutions. In [13] , Portilheiro and Vázquez considered the parabolic equation

,

with. They proved existence and uniqueness of viscosity solutions and derived the asymptotic behavior of the solutions for the Cauchy problem and the initial and Dirichlet problem with zero boundary conditions. In [14] , Portilheiro and Vázquez studied the nonlinear porous medium type equation involving the infinity Laplacian operator

By the density-to-pressure transformation, they transformed the Equation (4) into a new equation, then the existence, uniqueness and asymptotic behavior etc. were obtained.

In this article, we are interested in the parabolic version of the infinity Laplacian here. We think that Equation (1) is interesting, because it not only is degenerate, but also has many applications in image processing and optimal transportation etc. The parabolic equation involving infinity Laplacian operator has received a lot of attention in the last decade, notably due to its application to image processing, the main usage being in the reconstructions of damaged digital images [15] . For numerical purposes it has been necessary to consider also the evolution equation corresponding to the infinity Laplace operator. We prove the existence of viscosity solu- tions to the initial-Dirichlet problem by approximating procedure. The approximation process is introduced in [6] for the infinity Laplacian evolution and followed in [9] [13] [14] etc.

This paper is organized in the following order. In Section 2, we give the notations, definitions of viscosity solutions related to the Equation (1). In Section 3, we prove our main existence result by approximating pro- cedure.

2. Preliminaries

Throughout of this paper, we use the following notation: If, , denotes the lateral bouncary, the bottom boundary, and (the para- bolic boundary of). and denote the largest and the smallest of the eigenvalues to a symmetric matrix. denotes those functions which are twice differentiable in and once in.

In the following paper, we adopt the definition of viscosity solutions, (see for example [16] ).

Definition 2.1. Suppose that is upper semi-continuous. If for every and test function such that has a strict local maximum at point, that is and in a neighborhood of, there holds

then we say that is a viscosity sub-solution of (1).

Similarly, is lower semi-continuous. If for every and test function such that has a strict local minimum at point, there holds

then we say that is a viscosity super-solution of (1).

If is both a viscosity sub-solution and a viscosity super-solution, then we say that is a vis- cosity solution of (1).

3. Existence Theorem

In this section we will prove the existence of viscosity solutions to (1) with the initial and boundary data. The method we adopt is the approximation procedure introduced in [6] and used in [9] [13] [14] etc. The main existence result we obtain is.

Theorem 3.1. Let, where is a bounded domain, is continuous in, and let. Then there exists a function such that on and

in in the viscosity sense.

We use the approximate procedure, cf, [6] [9] [14] . We consider the approximating equations

where

with. For this equation with smooth initial and boundary data, the existence of a smooth solution is guaranteed by classical results in [17] . Our goal is to obtain a solution of (1) as a limit of these functions as. This amounts to proving uniform estimates for that are independent of. The estimates we require will be obtained by using the standard barrier method.

Theorem 3.2. (Boundary regularity at) Let, where is a bounded domain, is continuous in, and let. Suppose that is a smooth solution satisfying

Then there exists a constant depending on, and but independent of such that

.

Moreover, if is only continuous in (possibly discontinuous in) and bounded in, then the modulus of continuity of on (for small) can be estimated in terms of, and the modulus of continuity of in.

Proof. Step 1. Suppose first that and we consider the upper barrier function

,

where is to be determined. We have

if. Therefore is a super-solution.

Clearly, for all. Moreover, for and,

,

if, That is, on.

Thus, by the classical comparison principle, we obtain

for every. Similarly, by considering also the lower barrier function

,

we obtain the symmetric inequality, and hence the Lipschitz estimate

for and

.

Step 2. Suppose now that is only continuous in and let be its modulus of continuity. Let us fix

a point and. Let us consider the smooth functions

.

It is easy to check that on the parabolic boundary.

Thus if are the unique classical solutions to (7) with boundary and initial data, respectively, we have in by the classical comparison principle again. Since are smooth, we can use estimate (9) to conclude that

,

where depends on, and. Therefore,

with this inequality it is straightforward to complete the proof. □

The full Lipschitz estimate in time now follows easily with the aid of the comparison principle and the fact that the Equation (7) is translation invariant.

Corollary 3.3. (Lipschitz regularity in time) If is continuous in, and is as in

Theorem 3.2, then there exists a constant depending on, and but independent of

such that

for all and. Moreover, if is only continuous, then the modulus of continuity of on can be estimated in terms of, and the modulus of continuity of.

Proof. Let,. Then both u and ũ are smooth solutions to (7) in,

and hence if, we have

by the classical comparison principle and Theorem 3.2. This implies the Lipschitz estimate asserted above, and the proof for the case when is only continuous is analogous. □

Theorem 3.4. (Hölder regularity at the lateral boundary) Let, where is a bounded domain, is continuous in, and let. Suppose that is a smooth solution satisfying

Then for each, there exists a constant depending on, , , and but independent of and sufficiently small such that

,

for all and.

Proof. Step 1. For every and, let

,

where, are to be determined. Then a straightforward computation gives

If and, we have

.

Therefore

,

if.

We have shown that is a super-solution of (7).

Step 2. Let, where. We want to prove first on. Case 1. If, then

provided and.

Case 2. If, it is easy to see that is a super-solution of (7) in and on. Hence, we have

provided, and in the last inequality we have used the comparison principle.

Step 3. To prove on.

Case 1. If, then, and notice that since on the bottom of this cylinder,

if and.

Case 2. If, then. Using the comparison principle again, we have

if.

Step 4. In conclusion, we have shown that on, if we choose

,

.

Therefore, we have in by the comparison principle. In particular,

for. Using the lower barrier

,

we get the symmetric inequality. This finishes the proof. □

Due to the translation invariant of the equation and the comparison principle, we can extend the Hölder estimate to the interior of the domain, cf. [6] [14] etc.

Corollary 3.5. (Hölder regularity in space) Let, where is a bounded domain, is continuous in, and let. Suppose that a smooth solution satisfying

Then there exists, and constants, depending on , , and but independent of and sufficiently small such that

,

for all.

Proof. Step 1. For fixed, take a point and let. Define.

By Theorem 3.4 we have that on (noting that in this case or

). Hence for every by the comparison principle. This means that whenever or with, we have.

Step 2. When, using the comparison principle we get

.

This finishes the proof. □

The following theorem shows that one can obtain the Lipschitz estimate when one remove the Laplacian term from the equation, cf. [6] .

Theorem 3.1 follows now easily from Theorem 3.2 and 3.3 and the stability properties of viscosity solutions.

Proof. (Proof of Theorem 3.1) If and is the unique smooth solution to

Corollaries 3.3 and 3.5 and the comparison principle imply that the family of functions is equicon- tinuous and uniformly bounded. Therefore, up to a subsequence, as and is the unique viscosity solution to (7) by the stability properties of viscosity solutions.

The existence for a general continuous data follows by approximating the data by smooth functions and using Corollaries 3.3 and 3.5 and the stability properties of viscosity solutions again. □

Acknowledgements

The author would like to thank the anonymous referee for some valuable suggestions.

Support

This work is supported by the National Natural Science Foundation of China, No.11171153.

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