Global Continuation beyond Singularities on the Boundary for a Degenerate Diffusive Hamilton-Jacobi Equation
In this article, we are interested in the Dirichlet problem for parabolic viscous Hamilton-Jacobi Equations. It is well-known that the gradient of the solution may blow up in finite time on the boundary of the domain, preventing a classical extension of the solution past this singularity. This behavior comes from the fact that one cannot prescribe the Dirichlet boundary condition for all time and, in order to define a solution globally in time, one has to use “generalized boundary conditions” in the sense of viscosity solution. In this work, we treat the case when the diffusion operator is the p-Laplacian where the gradient dependence in the diffusion creates specific difficulties. In this framework, we obtain the existence and uniqueness of a continuous, global in time, viscosity solution. For this purpose, we prove a Strong Comparison Result between semi-continuous viscosity sub and super-solutions. Moreover, the asymptotic behavior of is analyzed through the study of the associated ergodic problem.
Key-words : Viscous Hamilton-Jacobi Equations, generalized Dirichlet problem, maximum principle, viscosity solutions, p-Laplacian.
AMS subject classifications : 35K10, 35K20, 49L25, 53C44 , 35B50, 35B05.
1 Introduction and Main Results
In this article we are interested in the following generalized Dirichlet problem for second-order degenerate parabolic partial differential equations
where , and are continuous functions satisfying the compatibility condition
Most of works devoted to this degenerate diffusive Hamilton-Jacobi equation concerned the case where , providing results on well-posedness, gradient estimates and asymptotic behavior of either classical or weak solutions in the sense of distributions (see [10, 1, 24, 22] and the references therein).
Some other works are concerned with the solvability of the Cauchy-Dirichlet problem. They proved that, under suitable assumptions on and , there exists a weak solution on some time interval , with the property that its gradient blows up on the boundary while the solution itself remains bounded. We refer the reader to   and  for the degenerate parabolic case and to  for the uniformly parabolic case. This singularity is a difficulty to extend the solution past . A natural question is then: Can we extend the weak solution past and in which sense ?
Recently, for the linear diffusion case (), Barles and Da Lio  showed that such gradient blow-up is related to a loss of boundary condition and address the problem through a viscosity solutions approach. They proved a ”Strong Comparison Result” (that is a comparison result between discontinuous viscosity sub and supersolutions) which allowed them to obtain the existence of a unique continuous, global in time viscosity solution of (1.1)–(1.3), the Dirichlet boundary condition being understood in the generalized sense of viscosity solution theory. They also provided an explicit expression of the solution of (1.1)–(1.3) in terms of a value function of some exit time control problem, which allows a simple explanation of the losses of boundary condition when it arises.
Our first result mainly extends the investigation of  to the degenerate diffusion case .
As it is classical in viscosity solutions theory, the proof of Theorem 1.1 relies on a Strong Comparison Result (SCR in short), the existence of the global solution being an almost immediate consequence of the Perron’s method introduced in the context of viscosity solutions by Ishii  (see also ).
The most important difficulties in the proof of Strong Comparison Results come from the formulation of the boundary condition in the viscosity sense, the discontinuity of the sub and the supersolution to be compared and the strong nonlinearity of the Hamiltonian term . A key argument in the proof of the SCR in  is the "cone condition" which is useful in the treatment of boundary points. Roughly speaking the ”cone condition” holds if at any point of the boundary , an usc subsolution satisfies where is a sequence of points of with the following properties
where is a positive constant.
Our approach is slightly different: instead of directly proving the ”cone condition” for any viscosity subsolution of (1.1)–(1.3) as it was done in , we use a combination of a regularity result for subsolutions of stationary problems, strongly inspired by the result of Capuzzo Dolcetta, Leoni and Porretta , together with a regularization by a sup-convolution in time. These arguments provide an approximation of the (a priori only usc) subsolution by a continuous subsolution, which automatically satisfies the “cone condition”, allowing to borrow the methods of  to conclude.
The generalisation of the regularity result of  is the following.
If is a locally bounded, usc viscosity subsolution of
where is an open subset of and is a positive constant, and if , then with .
Moreover, if is a bounded domain with a -boundary, then is bounded on and it can be extended as a -function on and
for some positive constant depending only on , and .
The regularity result of  was revisited in , where an interpretation was given in terms of state-constraint problems together with several possible applications. Our proof will rely on the arguments of .
A second motivation where such regularity results are useful, is the asymptotic behavior as of solutions of the evolution equation. For this purpose, one has first to study the ergodic (or additive eigenvalue) problem
associated to a state-constraint boundary condition on
We recall that, in this type of problems, both the solution and the constant (the ergodic constant) are unknown. First we have the following result.
The next step in the study of the asymptotic behavior would be to show that as where solves (1.9)–(1.10). The main difficulty to prove such more precise asymptotic behavior comes from the fact that (1.9)–(1.10) does not admit a unique solution ((1.9)–(1.10) is invariant by addition of constants). Such results were obtained recently in  for the uniformly elliptic case through the use of the Strong Comparison Principle (i.e. a result which allows to apply the Strong Maximum Principle to the difference of solutions) and the Lipschitz regularity of . But, for , such Strong Comparison Principle is not available since the equation is quasilinear and not semilinear. We recall that a Strong Maximum Principle is available for , see . Another difficulty comes from the proof of a strong comparison result for the steady problem in case of an operator that does not fulfill a monotonicity property, even if there exits a strict subsolution. Let us mention the works of [23, 7] for more results on the asymptotic behavior of global solutions.
Finally we point out that it was shown in  that the expected asymptotic behavior, namely , is not always true in the -case when the nonlinearity is sub quadratic in .
This article is organized as follows: in Section 2, we present the needed results on viscosity solutions for the stationary and evolution problems we consider; in particular, we analyze the losses of boundary conditions for subsolutions. In Section 3 we prove the Hölder regularity result of Theorem 1.2. In Section 4 we study the ergodic problem. Section 5 is devoted to the proof of Theorem 1.1 and the asymptotic behavior of solutions of the evolution equation.
2 Preliminaries and Analysis of Boundary Conditions
In this section we collect some preliminary properties of viscosity subsolutions (the boundary conditions being always understood in the viscosity sense) and we also formulate SCR under different forms, some of them being only useful as a step in the proof of the complete regularity result. These results are concerned with either problem (1.1)–(1.3) or the following two nonlinear elliptic problem
where , , and .
From now on, we assume that is a smooth domain with a -boundary. We define the distance from to by . For , we denote by
As a consequence of the regularity of , is a -function in a neighborhood of the boundary for all . We denote by a -function agreeing with in such that in . We also denote by the -function defined by in ; if , then is just the unit outward normal vector to at .
Our first result says that there is no loss of boundary conditions for the subsolutions, namely that the subsolutions satisfy the boundary condition in the classical sense.
We only give the proof for the time dependent problem, the proof for the stationnary problems being similar. We use a result of Da Lio [14, Corollary 6.2]. We denote by the space of real symmetric matrices. For , , and , we define the function by
so that the equation can be written as . From , we know that, if at some point , then the following conditions hold
But the first condition cannot hold since
and the right hand side is going to as since , and all terms converge to . ∎
Let us point out that the above computation shows that there is no competition between the nonlinear Hamiltonian term and the slow diffusion operator since they both produce positive contribution which prevent any loss of boundary conditions for the subsolution.
Next, we remark that there cannot be loss of initial condition.
Proof Fix and define for and the function by
This function attains a global maximum on at . Using the boundedness of , it is easy to see that, for any , as . Arguing as in , choosing sufficiently large depending on , we are left with and the two following possibilities
In either case, since , we get the desired result for letting and using the continuity of and . The argument for is similar.
Now we claim that under some assumptions (set out below), a SCR holds for semicontinuous viscosity sub-and supersolutions of (1.1)–(1.3) or (2.1) or (2.2). The proof being somehow technical we refer the reader to the appendice for a detailed proof of the following two propostions.
Proposition 2.2 (Parabolic SCR).
The stationary version of the SCR is used either in the proof of the -regularity or for solving the ergodic problem.
Proposition 2.3 (Elliptic SCR).
Assume that , and .
3 Hölder Regularity of Viscosity Subsolutions for the Degenerate Elliptic Problem
In this section we are going to prove that equation of type (1.7) enters into the general framework described in  which allows us to state that, if is a locally bounded, usc viscosity subsolution of (1.7), then is Hölder continuous with exponent . The key point is that the strong growth of the first order term balances the degeneracy of the second order term, providing a control on .
Now we are going to check the required hypotheses in .
H1. For , and , denoting the space of real valued symmetric matrices, define the function by
Then, for any with , in .
H2. There exists a super-solution up to the boundary such that , in and
for some .
Despite the construction of the functions is a rather easy adaptation of , we reproduce it for the sake of completeness and for the reader’s convenience. In order to build , we first build and then use the scale invariance of the equation. To do so, we borrow arguments from . For to be chosen later on and for , we consider the function
where on and we regularize it in by changing it into where is a smooth, non-decreasing and concave function such that is constant for and for . Obviously we have , in and is smooth in .
We first remark that can be written as
Therefore, in order to prove the claim, we are going to show that, for large enough, the bracket is positive and bounded away from and that remains large.
Computing the derivatives of in , we have
Using that for some and that , we have
Using that , with being , non-decreasing and concave, , we have
At this point, it is worth noticing that because of the properties of , the term is bounded.
These properties imply that, we can (almost) consider the two terms (in and in ) separately. Since , the term yields
By choosing large enough, we can have for any
On the other hand the term yields
We have to consider two cases: either and then , and ; hence the above quantity is given by
Recalling that , then for large enough we have for any
Now for , the quantity (3.2) coming from the -term is bounded and can be controlled by the -term. Hence, for any constant , choosing first large enough and then large enough, we have in
Next we set
It is easy to check that for , on .
H3. Comparison result. Let be any bounded usc viscosity subsolution of in then
We use the fact that is a strict super-solution up to the boundary and that it is a continuous function. It follows that the comparison is a direct consequence of Proposition 2.3.
Since the hypotheses are satisfied, we can apply Proposition 2.1 of  to obtain the regularity of subsolutions, both locally and globally with further assumptions on .
As far as the exponent is concerned, the value is the best one can expect in the assumption of the above theorem (see ).
It is well-known that the degeneracy of the -Laplacian is an an obstruction to the solvability of the Dirichlet problem in the classical sense. The presence of the strongly non-linear term with is another source of obstruction, even in the uniformly elliptic case since examples of boundary layers can occur [6, 20]. By the previous result, we know that every continuous solution to (1.7) is Hölder continuous up to the boundary. Hence, a necessary condition in order that the solution can attain continuously the boundary data is the existence of some such that
For the uniformly elliptic case , a more detailed study including several gradient bounds and applications can be found in .
As an application of the previous regularity result, we consider the generalized Dirichlet problem consisting in solving (2.2).
For the comparison part, Theorem 1.2 implies that is Hölder continuous, hence the comparison is a direct consequence of Proposition 2.3. Once noticed that and are respectively sub and super-solution, we can apply the Perron’s method with the version up to the boundary (see ). Since a solution is also a subsolution, the Hölder regularity is a direct consequence of Theorem 1.2.
4 The Ergodic Problem
4.1 Existence of the pair
In this part we study the existence of a pair for which is a viscosity solution of the state-constraints problem (1.9)-(1.10), to gather with the uniqueness of the ergodic constant . For this purpose, we introduce a -term in the equation, as it is classical, with the aim to let tend toward . This key step is described by the following Lemma.
Let and . For and , there exists a unique viscosity solution of the state constraint problem
Moreover there exists a constant such that, for all ,
For , we consider the following generalized Dirichlet problem
By Theorem 3.1, this problem admits a unique viscosity solution .
Indeed, on the one hand, it is easy to see that is a subsolution. On the other hand, borrowing arguments from , we claim that for some chosen large enough, is a supersolution of (4.1)-(4.2). Indeed, using that , we have
In where and , we have
Taking and such that
then we have