We consider viscosity solutions of the normalized p-Laplace equation
in , Ω being a domain in . Formally, the equation reads
In the linear case , we have the heat equation , and also for , the equation reduces to the heat equation . At the limit , we obtain the equation for motion by mean curvature. We aim at showing that the time derivative exists in the Sobolev sense and belongs to . We also study the second derivatives .
There has been some recent interest in connection with stochastic game theory, where the equation appears, cf. . From our point of view, the work  is of actual interest, because there it is shown that the time derivative of the viscosity solutions exists and is locally bounded, provided that the lateral boundary values are smooth. Thus, the boundary values control the time regularity. If no such assumptions about the behaviour at the lateral boundary are made, a conclusion like is in doubt. Our main result is the following, where we unfortunately have to restrict p.
Suppose that is a viscosity solution of the normalized p-Laplace equation in . If , then the Sobolev derivatives and exist and belong to .
We emphasize that no assumptions on the boundary values are made for this interior estimate. Our method of proof is based on a verification of the identity
where we have to prove that the function U, which is the right-hand side of equation (1.1), belongs to . Thus, the second spatial derivatives are crucial (local boundedness of was proven in [3, 2] and interior Hölder estimates for the gradient in ). The elliptic case has been studied in .
In the range , one can bypass the question of second derivatives.
Suppose that is a viscosity solution of the normalized p-Laplace equation in . If , then the Sobolev derivative exists and belongs to .
To avoid the problem of vanishing gradient, we first study the regularized equation
Here the classical parabolic regularity theory is applicable. The equation was studied by Does in , where an estimate of the gradient was found with Bernstein’s method. We shall prove a maximum principle for the gradient. Further, we differentiate equation (1.2) with respect to the space variables and derive estimates for , which are passed over to the solution u of (1.1).
Analogous results seem to be possible to reach through the Cordes condition. This also restricts the range of valid exponents p. We have refrained from this approach, mainly since the absence of zero (lateral) boundary values produces many undesired terms to estimate. Finally, we mention that the limits and in Theorem 1.1 are evidently an artifact of the method. It would be interesting to know whether the theorem is valid in the whole range . In any case, our method is not capable to reach all exponents.
The gradient of a function is
and its Hessian matrix is
We shall, occasionally, use the abbreviation
for partial derivatives. Young’s inequality
is often referred to. Finally, the summation convention is used when convenient.
The normalized p-Laplace equation is not in divergence form. Thus, the concept of weak solutions with test functions under the integral sign is problematic. Fortunately, the modern concept of viscosity solutions works well. The existence and uniqueness of viscosity solutions of the normalized p-Laplace equation was established in . We recall the definition.
We say that an upper semi-continuous function u is a viscosity subsolution of equation (1.1) if for all , we have
at any interior point where attains a local maximum, provided . Further, at any interior point where attains a local maximum and , we require
for some , with .
We say that a lower semi-continuous function u is a viscosity supersolution of equation (1.1) if for all , we have
at any interior point where attains a local minimum, provided . Further, at any interior point where attains a local minimum and , we require
for some , with .
A continuous function u is a viscosity solution if it is both a viscosity subsolution and a viscosity supersolution.
For a detailed discussion on the definition at critical points, we refer to . The reason behind the choice of is given in [5, Section 2]. The viscosity solutions of equation (1.2) are defined in a similar manner, except that now is not a problem.
Maximum principle for the gradient.
In order to estimate the time derivative, we need bounds on the second derivatives of (and also on its gradient). If we first assume that is on the parabolic boundary , we get bounds on the gradient in all of . This follows from the following maximum principle.
Proposition 2.4 (Maximum principle).
Let be a solution of equation (1.2). If , then
With some modifications, a proof can be extracted from . We give a direct proof. To this end, consider
To find the partial differential equation satisfied by , we calculate1
Writing equation (1.1) in the form
Rearranging and using
we arrive at the following differential equation for :
Suppose that has an interior maximum point at . At this point, , otherwise we would have in , in which case there is nothing to prove. By the infinitesimal calculus,
where we have included the case . Further, the matrix is negative semidefinite. Using equation (2.1) and noting that and , we get, at ,
since the matrix A, with elements , is positive semidefinite. To avoid the contradiction , w must attain its maximum on the parabolic boundary.
Hence, for any , we have
We finish the proof by sending . ∎
With no assumptions for on the parabolic boundary, we need a stronger result, taken from [3, p. 381].
Let be a solution of equation (1.2), with . Then
Note that no condition on the lateral boundary was used. By continuity,
for and . The estimate
follows. (Here one can pass to the limit as .)
The proof of the lemma below, a simple special case of the Miranda–Talenti lemma, can be found for smooth functions in [4, p. 308]. If f is not smooth, we perform a strictly interior approximation, so that no boundary integrals appear (which is possible since ).
Lemma 2.6 (Miranda–Talenti).
Let and . Then
Then uniformly on compact subsets of .
By Theorem 2.5, we can use Ascoli’s theorem to extract a convergent subsequence converging locally uniformly to some continuous function, namely, . We claim that v is a viscosity solution of equation (1.1). The lemma then follows by uniqueness.
We demonstrate that v is a viscosity subsolution. (A symmetric proof shows that v is a viscosity supersolution.) Assume that attains a strict local maximum at . Since locally uniformly, there are points
such that attains a local maximum at . If , then for all small enough, and at , we have
Letting , we see that v satisfies Definition 2.3 when . If , let
Since , there is a subsequence such that when for some , with . Passing to the limit in equation (3.1), we see that v is a viscosity subsolution. ∎
Our proof of Theorem 1.1 consists in showing that the second derivatives belong locally to with a bound independent of ϵ. Once this is established, we see that
Hence, for any bounded subdomain ,
with C independent of ϵ. By this uniform bound, there exists a subsequence such that, as ,
In particular, this means that and for any , we have
This shows that the Sobolev derivative exists and, since the previous equation holds for any subdomain , we conclude that . To complete the proof of Theorem 1.1, it remains to establish the missing local bound of uniformly in ϵ.
4 The differentiated equation
We shall derive a fundamental identity. Let
Differentiating equation (1.2) with respect to the variable , we obtain
Take , with . Multiply both sides of the equation by and sum j from 1 to n. Integrate over , using integration by parts and keeping in mind that ξ is compactly supported in , to obtain
Writing out the derivatives gives the fundamental formula
In the next section we shall bound the main term uniformly with respect to ϵ.
5 Estimate of the second derivatives
We shall provide an estimate of the main term . First, we record the elementary inequality
One dimension. As an exercise, we show that in this case, the second derivatives are locally bounded in for any . In one dimension, equation (1.1) reads
We absorb the terms and , using Young’s inequality and inequality (5.1). For any ,
Applying Theorem 2.5 we see that the right-hand side is bounded by a constant independent of . We have
It follows that locally for any .
General n. We assume for the moment that . We rewrite the term involving the Laplacian as
Upon this rewriting, the term disappears from the equation. We focus our attention on the term involving . By Lemma 2.6,
Differentiating, we see that
It follows that
where depends only linearly on the second derivatives :
By Young’s inequality, we obtain
Inserting this into the main equation gives
All terms containing can be absorbed by the new main term . To this end, we use Young’s inequality with a small parameter to balance the terms.2 For term , we have
Similarly, for term ,
Using similar inequalities for the term involving and choosing the parameters small enough in Young’s inequality, we find,
A symmetric proof when shows that equation (5.2) holds when
6 The case
In this section, we give a proof of Theorem 1.2. To this end, let , with . We claim that
where the supremum norm of is taken locally, over the support of ξ. Here, is the solution of the regularized equation (1.2). This is enough to complete the proof of Theorem 1.2, in virtue of Theorem 2.5.
Multiplying the regularized equation (1.2) by yields
The integral of the divergence term vanishes by Gauss’s theorem and, upon integration, we have
The first integral on the right-hand side can be absorbed by the left-hand side by choosing in
For the last term, the decisive observation is that
We use this in the last integral on the right-hand side to obtain
To sum up, we have now the final estimate
So far, our calculations are valid in the full range . For , we have
We thank Amal Attouchi for valuable help with a proof.
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About the article
Published Online: 2018-07-12
Published in Print: 2019-03-01
Funding Source: Norges Forskningsråd
Award identifier / Grant number: 250070
Supported by the Norwegian Research Council (grant 250070).
Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 7–15, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0091.
© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0