Gradient estimates for an orthotropic nonlinear diffusion equation

1.1 Aim of the paper

Let Ω ⊂ ℝ N be an open bounded set and I ⊂ ℝ 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 [27] by Vishik. Equation (1.1) also explicitly appears in the monographs [21], [23, Example 4.A, Chapter III] and [29, Example 30.8], among others.

In this paper, we will focus on the case p ≥ 2 . 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 F : ℝ N → ℝ 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 [13], 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 ∇ ⁡ u (i.e. boundedness and continuity), the situation becomes fairly more complicated. Let us start from (1.2). DiBenedetto and Friedman [14] 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 div ⁢ ∇ ⁡ F is restricted to a single point, since the Hessian D 2 ⁢ F behaves as in the model case (1.2)

where the elliptic character is lost only for z = 0 . 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 z ∈ ℝ N such that one component z i 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 L ∞ bound on ∇ ⁡ u 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 [7], 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 [11] 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 ∇ ⁡ u refers to the spatial variables, i.e. ∇ ⁡ u = ( u x 1 , … , u x N ) .

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 F ε is a smooth uniformly convex approximation of the orthotropic function (1.3). By the classical regularity theory, the maps u ε are regular enough to justify all the calculations below. The goal is to establish a local uniform Lipschitz estimate on u ε , which does not depend on the regularization parameter ε. Finally, we let ε go to 0 and prove that the family u ε 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 u ε . In order to simplify the presentation, we drop the index ε both for F ε and u ε . 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 x j , 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 u x j with a non-negative convex function h is a subsolution of this equation. Accordingly, the map h ⁢ ( u x j ) satisfies the Caccioppoli inequality which is naturally attached to (1.5) (see Lemma 3.1 below). If I = ( T 0 , T 1 ) and τ ∈ ( T 0 , T 1 ) , this reads as follows:

(1.6)

Here, the maps χ ∈ C 0 ∞ ⁢ ( I ) and η ∈ C 0 ∞ ⁢ ( Ω ) are non-negative cut-off functions in the time and space variables, respectively. We have used the following expedient notation: given f ∈ L 1 ⁢ ( I × Ω ) ,

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 h ⁢ ( u x j ) ; that is, a sort of reverse Poincaré inequality where the Sobolev norm of h ⁢ ( u x j ) is controlled by the L 2 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 | ∇ ⁡ u | p - 2 is recombined with the subsolution h ⁢ ( u x j ) 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 u x j . This is nowadays a standard technique in the field; for the elliptic case, it goes back to the pioneering works by Ural’tseva [26] and Uhlenbeck [25].

As we explained above, due to the severe degeneracy of D 2 ⁢ F 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 [3] and then fruitfully exploited in [5].

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 D 2 ⁢ F 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.

Let us come back for one instant to the standard Caccioppoli inequality (1.6). It follows from (1.5) by taking φ = h ⁢ h ′ ⁢ ( u x j ) ⁢ χ ⁢ η 2 . In particular, the parabolic term is given by

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 h 2 ⁢ ( u x j ) as in the elliptic framework.

For the weird Caccioppoli inequalities, the test function is now φ = u x j ⁢ Φ ′ ⁢ ( u x j 2 ) ⁢ Ψ ⁢ ( u x k 2 ) ⁢ χ ⁢ η 2 , for some 1 ≤ j , k ≤ N . 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 Φ ⁢ ( u x j 2 ) , since it would affect the factor Ψ ⁢ ( u x k 2 ) . 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 u x j and u x k .

Basically, we merge together two weird Caccioppoli inequalities, where the spatial variables x j and x k 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 L ∞ estimate on ∇ ⁡ u .

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 u ε .

2.1 Local solutions

Let Ω ⊂ ℝ N be an open bounded set and I ⊂ ℝ an open bounded interval. Fix p > 2 and take 𝒜 : ℝ N → ℝ N a continuous function such that

and

We say that u ∈ L loc p ⁢ ( I ; W loc 1 , p ⁢ ( Ω ) ) is a local weak solution of the quasilinear diffusion equation

if for every φ ∈ C 0 ∞ ⁢ ( I × Ω ) we have

2.2 Steklov averages

Throughout the paper, we denote by T 0 < T 1 the endpoints of the time interval I. Let v ∈ L loc 1 ⁢ ( I × Ω ) . For every 0 < σ < T 1 - T 0 , we define its so-called

1. forward Steklov average

   v σ + ⁢ ( t , x ) = ⨍ t t + σ v ⁢ ( τ , x ) ⁢ 𝑑 τ   for every  ⁢ ( t , x ) ∈ ( T 0 , T 1 - σ ) × Ω ,

2. backward Steklov average

   v σ - ⁢ ( t , x ) = ⨍ t - σ t v ⁢ ( τ , x ) ⁢ 𝑑 τ   for every  ⁢ ( t , x ) ∈ ( T 0 + σ , T 1 ) × Ω .

We shall use some standard properties of the Steklov averages. Let 0 < σ < T 1 - T 0 and ψ ∈ L ∞ ⁢ ( I × Ω ) such that ψ is compactly supported in ( T 0 , T 1 - σ ) × Ω . We extend ψ by 0 on ( ℝ ∖ I ) × Ω , so that ψ σ - is well defined on I × Ω and compactly supported therein. By the Fubini theorem, we have

Moreover, if v ∈ L loc q ⁢ ( I × Ω ) for some 1 ≤ q < ∞ , then v σ + converges to v in L loc q ⁢ ( I × Ω ) , 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:

3.1 An approximating equation

We denote by

and for every ε ∈ ( 0 , 1 ) , we consider the convex function

We consider a local weak solution u ε ∈ L loc p ⁢ ( I ; W loc 1 , p ⁢ ( Ω ) ) of equation (2.1) with the choice

This means that u ε verifies

for every φ ∈ C 0 ∞ ⁢ ( I × Ω ) . Observe that the map F ε belongs to C 2 ⁢ ( ℝ N ) and satisfies

Hence, one can rely on the classical regularity theory for quasilinear parabolic equations, see e.g. [13, Theorem 5.1, Chapter VIII] and [1, Lemma 3.1], to get

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 x j , 1 ≤ j ≤ N .

Fix 0 < σ < T 1 - T 0 . Let ψ ∈ C 0 ∞ ⁢ ( ( T 0 , T 1 - σ ) × Ω ) . As already observed, the backward Steklov average

is compactly supported in I × Ω . We can thus insert it into (3.2):

Since ( ψ σ - ) t = ( ψ t ) σ - , equation (2.2) implies that

One thus gets

for every ψ ∈ C 0 ∞ ⁢ ( ( T 0 , T 1 - σ ) × Ω ) .

Let j ∈ { 1 , … , N } and φ ∈ C 0 ∞ ⁢ ( ( T 0 , T 1 - σ ) × Ω ) . The map

belongs to L loc p ⁢ ( ( T 0 , T 1 - σ ) ; W loc 1 , p ⁢ ( Ω ) ) and ( ( u σ + ) t ) x j = ( ( u x j ) σ + ) t . By derivation under the integral sign, one also has

We insert ψ = φ x j 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 F ∈ C 2 ⁢ ( ℝ N ) and ∇ ⁡ u ∈ L loc ∞ ⁢ ( I × Ω ) ∩ L loc 2 ⁢ ( I ; W loc 1 , 2 ⁢ ( Ω ) ) , one has

We can thus appeal to a density argument to get that (3.5) remains true for every φ ∈ L 2 ⁢ ( I ; W 1 , 2 ⁢ ( Ω ) ) , with compact support in ( T 0 , T 1 - σ ) × Ω .

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 W 1 , 2 estimate on h ⁢ ( u x j ) , where h is any smooth convex function.