We consider a quasilinear degenerate parabolic equation driven by the orthotropic p-Laplacian. We prove that local weak solutions are locally Lipschitz continuous in the spatial variable, uniformly in time.
1.1 Aim of the paper
Let be an open bounded set and an open bounded interval. We study the gradient regularity of local weak solutions to the following parabolic equation:
Evolution equations of this type have been studied since the 1960s, especially by the Soviet school, see for example the paper  by Vishik. Equation (1.1) also explicitly appears in the monographs , [23, Example 4.A, Chapter III] and [29, Example 30.8], among others.
In this paper, we will focus on the case . We first observe that (1.1) looks quite similar to the more familiar one
which involves the p-Laplace operator
Indeed, both parabolic equations are particular instances of equations of the type
with a convex function which satisfies the structural conditions
Then the basic regularity theory equally applies to both (1.1) and (1.2). The standard reference in the field is DiBenedetto’s monograph , where one can find boundedness results for the solution u (see [13, Chapter V]), Hölder continuous estimates for u (see [13, Chapter III]), as well as Harnack inequality for positive solutions (see [13, Chapter VI]). At a technical level, there is no distinction to be made between (1.1) and (1.2).
In contrast, when coming to the regularity of (i.e. boundedness and continuity), the situation becomes fairly more complicated. Let us start from (1.2). DiBenedetto and Friedman  have proved that the gradients of solutions to this equation are bounded. This is the starting point to obtain the continuity of the gradients for any
We refer again to DiBenedetto’s book for a comprehensive collection of results on the subject, notably to [13, Chapter VIII]. Since then, there has been a growing literature concerning the regularity for nonlinear, possibly degenerate or singular, parabolic equations (or systems), the main model of which is given by the evolutionary p-Laplacian equation (1.2). Without any attempt to completeness, we can just mention some classical references [15, 12, 28, 8, 9], up to the most recent contributions on the subject, given by [2, 19, 18], among others.
However, none of these results apply to our equation (1.1). Indeed, all of them rely on the fact that the loss of ellipticity of the operator is restricted to a single point, since the Hessian behaves as in the model case (1.2)
where the elliptic character is lost only for . Such a property dramatically breaks down for our equation (1.1). Indeed, in this case, the function F has the following orthotropic structure:
The Hessian matrix of F now degenerates on an unbounded set, namely the set of those such that one component is 0. As a consequence, the aforementioned references do not provide any regularity results for the gradients of the solutions.
The main goal of the present paper is to prove the bound on for our equation (1.1), thus extending the result by DiBenedetto and Friedman to this more degenerate setting. In order to do this, we will need to adapt to the parabolic setting the machinery that we developed in [3, 4, 5, 6] and , for degenerate equations with orthotropic structure. Indeed, the operator
that we called orthotropic p-Laplacian, is the prominent example of this kind of equations. We also refer to  for an approach to this operator, based on viscosity techniques.
1.2 Main result
In this paper, we establish the following regularity result which can be seen as the parabolic counterpart of our previous result [5, Theorem 1.1] for the elliptic case. In the statement below, the notation refers to the spatial variables, i.e. .
Let and let be a local weak solution of (1.1). Then . More precisely, for every parabolic cube
and every , we have
for a constant .
Remark 1.1 (Scalings).
We observe that equation (1.1) is invariant with respect to the “horizontal” and “vertical” scale changes
for every . Then it is easily seen that the a priori estimate (1.4) is invariant with respect to these scale changes. We point out that such estimate is the exact analogue of that for the evolutionary p-Laplacian, see [13, Theorem 5.1, Chapter VIII]. Occasionally, in the paper we will work with anisotropic parabolic cubes of the type
The choice of cubes of this type could be loosely justified by a dimensional analysis of the equation. Indeed, by considering the quantity u as dimensionless and using the family of scalings
we get the relation
However, as is well known, estimates on cubes of the type are too restrictive when looking at estimates for . Indeed, in light of the so-called intrinsic geometry, it is much more important to work with local estimates on cubes , where the time scale τ is adapted to the solution itself: roughly speaking, we can take
This explains the importance of having (1.4) with two independent scales R and τ. We refer to [13, Chapter VIII] for a description of the method of intrinsic scalings, where these heuristics are clarified.
Remark 1.2 (Case ).
When , the orthotropic parabolic equation (1.1) boils down to the standard heat equation, for which solutions are well known to be smooth. For this reason, in our statement we restrict our attention to the case . However, we point out that by making the choice and taking the limit as p goes to 2 in (1.4), we formally end up with the classical gradient estimate for solutions of the heat equation
In light of the previous remark, in the case the relation
is now the natural one.
As for the singular case , this is somehow simpler than its degenerate counterpart. In this case, the local Lipschitz regularity of solutions to (1.1) can be directly inferred from [22, Theorem 1], under the restriction
Indeed, the result of  covers (among others) the case of parabolic equations of the form
under the following assumptions on the convex function F:
It is not difficult to see that the orthotropic function (1.3) for matches both requirements. Indeed, observe that
thanks to the fact that . Thus in the subquadratic case, the orthotropic structure helps more than it hurts, in a sense.
Remark 1.3 (Anisotropic diffusion).
We conclude this part by observing that, more generally, one could consider the following parabolic equation:
which still has an orthotropic structure. Now we have a whole set of exponents , one for each coordinate direction. We cite the paper , where some global Lipschitz regularity results are proven for solutions of the relevant Cauchy–Dirichlet problem, under appropriate regularity assumptions on the data. We point out that in light of their global nature, for such results are not comparable to ours. We also refer to  for a sophisticated Harnack inequality for positive local weak solutions, as well as for some further references on the problem. Finally, the very recent paper  contains a thorough study of the Cauchy problem in the case , together with some regularity results.
However, as for the counterpart of our Main Theorem for local solutions of this equation, this is still an open problem, to the best of our knowledge.
1.3 Technical aspects of the proof
The core of the proof of the Main Theorem is an a priori Lipschitz estimate for smooth solutions of the orthotropic parabolic equation, see Proposition 4.1 below. More precisely, we introduce the regularized problem
where is a smooth uniformly convex approximation of the orthotropic function (1.3). By the classical regularity theory, the maps are regular enough to justify all the calculations below. The goal is to establish a local uniform Lipschitz estimate on , which does not depend on the regularization parameter ε. Finally, we let ε go to 0 and prove that the family converges to the original solution u. This allows to obtain the Lipschitz estimate for u itself.
In the subsequent part of this subsection, we emphasize the main difficulties to get such a Lipschitz estimate on . In order to simplify the presentation, we drop the index ε both for and . The strategy is apparently quite classical: we rely on a Moser iterative scheme of reverse Hölder’s inequalities, resulting from the interplay between Caccioppoli estimates and the Sobolev embeddings.
To be more specific, we first differentiate the equation with respect to a spatial variable , so as to get the equation solved by the j-th component of the gradient. This is given by
More generally, the composition of the component with a non-negative convex function h is a subsolution of this equation. Accordingly, the map satisfies the Caccioppoli inequality which is naturally attached to (1.5) (see Lemma 3.1 below). If and , this reads as follows:
Here, the maps and are non-negative cut-off functions in the time and space variables, respectively. We have used the following expedient notation: given ,
When F is a uniformly elliptic integrand, in the sense that
one can easily obtain from (1.6) a crucial “unnatural” feature of the subsolution ; that is, a sort of reverse Poincaré inequality where the Sobolev norm of is controlled by the norm of the subsolution itself. In conjunction with the Sobolev inequality, this is the cornerstone which eventually leads to the classical version of the Moser iterative scheme. It should be noticed that this strategy still works even in the degenerate case, provided the Hessian behaves like
as for the evolutionary p-Laplace equation (1.2). It is sufficient to use the “absorption of degeneracy” trick, where the degenerate weight is recombined with the subsolution by means of simple algebraic manipulations. This still permits to infer from (1.6) a control on the Sobolev norm of a suitable convex function of . This is nowadays a standard technique in the field; for the elliptic case, it goes back to the pioneering works by Ural’tseva  and Uhlenbeck .
As we explained above, due to the severe degeneracy of in our orthotropic situation, it is not possible to follow the same path. In order to rely on such an absorption trick, we have to go through a tour de force and to introduce a new family of weird Caccioppoli inequalities (see Lemma 3.2 below). These are the parabolic counterparts of a corresponding estimate introduced in the elliptic setting in  and then fruitfully exploited in .
The crucial idea is to mix together the components of the gradient with respect to 2 orthogonal directions. This compensates the lack of ellipticity of and allows to rely on the Sobolev embeddings in the iterative scheme. We do not detail these Caccioppoli-type estimates here, but instead explain the main additional difficulties with respect to the elliptic framework.
Then an integration by parts yields
The latter yields the “time slice” term on the left-hand side of (1.6). This way, the time derivative is transferred to χ and one can handle the factor as in the elliptic framework.
For the weird Caccioppoli inequalities, the test function is now , for some . The corresponding parabolic term becomes
In contrast to the previous situation, we cannot perform an integration by parts to get rid of the time derivative on , since it would affect the factor . In order to overcome this difficulty, which does not arise in the elliptic setting, we need a new approach, aimed at “symmetrizing” the above quantity containing and .
Basically, we merge together two weird Caccioppoli inequalities, where the spatial variables and play symmetric roles. More specifically, we insert into (1.5) the test functions
and then add the two resulting inequalities. The parabolic term is now replaced by the following quantity:
This allows to integrate by parts and transfer the time derivative on the test function. It turns out that by a suitable adaptation of the arguments that we used in the elliptic case, one can incorporate this new term in the iterative Moser scheme. This finally leads to the desired local estimate on .
1.4 Plan of the paper
The paper is organized as follows: after collecting the basic terminology and some preliminaries on Steklov averages in Section 2, we present in Section 3 the proofs of the new Caccioppoli inequalities in the parabolic setting. We detail the iterative Moser scheme in Section 4 and finally establish the Main Theorem in Section 5, by transferring to the original solution u the a priori estimates obtained on the approximating solutions .
2.1 Local solutions
Let be an open bounded set and an open bounded interval. Fix and take a continuous function such that
We say that is a local weak solution of the quasilinear diffusion equation
if for every we have
2.2 Steklov averages
Throughout the paper, we denote by the endpoints of the time interval I. Let . For every , we define its so-called
forward Steklov average
backward Steklov average
We shall use some standard properties of the Steklov averages. Let and such that ψ is compactly supported in . We extend ψ by 0 on , so that is well defined on and compactly supported therein. By the Fubini theorem, we have
Moreover, if for some , then converges to v in , as σ goes to 0, see e.g. [13, Chapter I, Lemma 3.2].
Finally, we can derive from (2.2) the following regularity properties of the Steklov averages:
Let . Then for every ,
the map belongs to and(2.3)
if one further assumes that , then and(2.4)
Fix . Let . Then by (2.2),
By an obvious change of variables, this yields
which gives the desired identity (2.3).
In the last equality, we have derived under the integral sign the smooth function ψ. Hence, by integrating by parts the last integral and using (2.2) again, one gets
from which (2.4) follows. ∎
3 Energy estimates for a regularized equation
3.1 An approximating equation
We denote by
and for every , we consider the convex function
We consider a local weak solution of equation (2.1) with the choice
This means that verifies
for every . Observe that the map belongs to and satisfies
In the following computations, we delete the index ε both for u and F.
3.2 An equation for the spatial gradient
In order to establish a Lipschitz bound on our solution u, we need to differentiate (3.2) with respect to the spatial variables , .
Fix . Let . As already observed, the backward Steklov average
is compactly supported in . We can thus insert it into (3.2):
Since , equation (2.2) implies that
One thus gets
for every .
Let and . The map
belongs to and . By derivation under the integral sign, one also has
We insert in equation (3.4). An integration by parts in the spatial variable leads to
Finally, using (2.2) in the second term, one gets
We observe that, since and , one has
We can thus appeal to a density argument to get that (3.5) remains true for every , with compact support in .
3.3 Caccioppoli-type inequalities
As explained in the introduction, the first technical tool in the proof of the Lipschitz bound of u is the following Caccioppoli inequality which provides a estimate on , where h is any smooth convex function.
Lemma 3.1 (Standard Caccioppoli inequality).
Let and be two non-negative functions, with χ non-decreasing. Let be a convex non-negative function. Then, for almost every and every , we have
We insert into (3.5) the test function
which has compact support in and belongs to . By (3.7),
We use the above identity to infer
We then perform an integration by parts with respect to the time variable in the first term
We now want to take the limit as σ goes to 0. Let such that η is compactly supported in . Since , we have
Moreover, we know that which guarantees that there exists such that for every ,
It then follows from the Dominated Convergence Theorem that
Next, by recalling the choice of φ above, we have
By Lemma 2.1, we know that
This implies that converges to in . Hence, a similar argument to the one leading to (3.8) implies that
Finally, by using that , we can infer that
It follows that
Up to now, we have thus proved
We now choose ζ as follows. Let be as in the statement. Given and such that , we define
We then insert
into (3.9). Then, for almost every , we can let δ go to 0 and obtain
Since χ does not depend on the spatial variable, we have
Since the second term is non-negative, by dropping it, we get from (3.11)
In order to estimate the last term, we use the Cauchy–Schwarz inequality
A further application of Young inequality leads to
In this way, the integral containing can be absorbed on the left-hand side. Let us finally observe that we can remove the assumption on the function h, by a standard approximation argument. ∎
We next establish the key tool for the proof of our main result, namely a Caccioppoli-type inequality, where two different partial derivatives and come into play.
Lemma 3.2 (Weird Caccioppoli inequality).
Let and be two non-negative functions, with χ non-decreasing. Let and be two non-decreasing and non-negative convex functions. Then, for almost every , every and every , we have
It is convenient to divide the proof into two steps.
Step 1: An identity involving and . We first assume that Φ and Ψ are two non-decreasing and non-negative convex functions. We fix . Given and , we consider (3.5) with the index j and for every , we insert the test function
Symmetrically, we consider (3.5) with the index k and insert the test function
The functions φ and are compactly supported in and belong to . Thus they are admissible test functions. We observe that