On the local behavior of local weak solutions to some singular anisotropic elliptic equations

1.1 Parabolic approach

To introduce our approach, we present an alternative proof of the mean value theorem for solutions to Laplace equation. This brief and modest scheme will highlight the essence of our method, that is, conceived to obtain classical properties of some elliptic equations through a parabolic approach. Let us consider the Laplace equation in an open bounded set Ω ⊂ ⊂ R N ,

Let x o ∈ Ω be a Lebesgue point for u , and let B 2 r ( x o ) be the ball of radius 2 r and center x o . Now for 0 < t < r such that B 2 r ( x o ) ⊂ Ω , consider the test function ϕ ( t , x ) = ( t 2 − ∣ x − x o ∣ 2 ) + , to obtain the integral equality

By Green’s formula, this is equivalent to

having used that n = ( x − x o ) / ∣ x − x o ∣ is the normal unit vector to ∂ B t ( x o ) and being d ℋ N − 1 the Hausdorff ( N − 1 ) -dimensional measure. Now, last display can be rewritten as

Finally, we integrate along t ∈ ( 0 , r ) and we use Lebesgue’s theorem to obtain

with ω N = ∣ B 1 ∣ and ∣ B r ∣ = ω N r − N . This implies the mean value property

This point-wise control given in integral average can be used in turn to derive very strong regularity properties of the solutions. We will undergo a similar strategy for solutions to (1.1), by taking into account the degeneracies and singularities that are typical of anisotropic equations.

1.2 Definitions and main results

Let Ω ⊂ R N be an open bounded set with N ≥ 2 , and let us denote with ∂ i the i th partial weak derivative. For 1 < p < 2 and 1 ≤ s ≤ N − 1 , we consider the elliptic partial differential equation

where the Caratheodory^[1] functions A i ( x , u , ξ ) : Ω × R × R N → R are subject to the following structure conditions for almost every x ∈ Ω ,

where C 1 , C 2 > 0 , C ≥ 0 are given constants that we will always refer to as the data. A function u ∈ L loc ∞ ( Ω ) ∩ W loc 1 , [ 2 , p ] ( Ω ) , where

is called a local weak solution to (1.2)–(1.3) if for each compact set K ⊂ ⊂ Ω it satisfies

Throughout this study, we will suppose that truncations ± ( u − k ) ± of local weak solutions to (1.2)–(1.3) preserve the property of being sub-solutions: for any k ∈ R , every compact subset K ⊂ Ω , and ψ ∈ W o 1 , [ 2 , p ] ( K ) we have

Properties of anisotropic Sobolev spaces have first been investigated in [18,22,29], and boundedness of local weak solutions has been first considered in [19] and refined in [14]. Limit growth conditions have been investigated in [16] and then refined in [8,9] in great generality. Henceforth, it is a well-known fact in the literature that local weak solutions to our equation (1.2) are bounded, provided p max ≤ N p ¯ / ( N − p ¯ ) , being p ¯ = N ( ∑ i = 1 N ( p i ) − 1 ) − 1 the harmonic mean. We consider the prototype equation to (1.2) as a special case of the full anisotropic analogue

with p i = 2 for i = 1 , … , s and 1 < p i = p < 2 on the remaining components. This last equation suffers heavily from the combined effect of singular and degenerate behavior, even when for instance all p i s are greater than two. This is because the natural intrinsically scaled geometry of the equation that maintains invariant the volume ∣ K ∣ = ρ N can be shaped on anisotropic cubes as

where M is a number depending on the solution u itself (indeed the epithet intrinsic) that vanishes as soon as u vanishes. Therefore, when M approaches zero, for those directions whose index satisfies p i > p ¯ the anisotropic cube K shrinks to a vanishing measure, while for the remaining ones it stretches to infinity. For a detailed description of this geometry and its derivation through self-similarity we refer to [5], where the evolutionary, fully anisotropic prototype equation is considered (see also [15] for the singular case).

We state our two main results hereafter. The first one is a result of the local Hölder continuity.

Next we fix some geometrical notations and conventions. For a point x o ∈ Ω , let us denote it by x o = ( x o ′ , x o ″ ) where x o ′ ∈ R s and x o ″ ∈ R N − s . Let θ , ρ > 0 be two parameters and define the polydisc

We will say Q θ , ρ is an intrinsic polydisc when θ depends on the solution u itself. We will call first s variables the nondegenerate variables and last ( N − s ) ones singular variables. Using this geometry we state our main result, an intrinsic form of Harnack inequality.

Condition (1.9) expresses the range of exponents available for the result to hold relatively to the weighted effect of singular and nondegenerate operators into play. When s decreases the range becomes tighter to the parabolic isotropic range for Harnack inequality to hold, i.e., 2 N / ( N + 1 ) < p < 2 . But when s increases the effect of regularization is stronger and this interval expands until it reaches 1 < p < 2 .

Theorems 1.1 and 1.2 are consequences of the following ones, which are worth of interest on their own.

We prove indeed the following shrinking property, which is typical of both singular parabolic equations ([10] Lemma 5.1 Chap. IV) and isotropic elliptic equations ([11] Prop 5.1 Chap X).

The aforementioned theorem provides also an important expansion of positivity along the singular variables, conceived and modeled in a similar fashion than in ([13], Theorem 2.3).

Another fundamental tool for our analysis of local regularity is the following integral estimate, which can be seen as an Harnack estimate within the L 1 − L ∞ topology, and is typical of singular parabolic equations (see for instance [10], Prop. 4.1 Chap VII).

1.3 Novelty and significance

Considering fully anisotropic equations as (1.1), a standard statement of regularity requires a bound on the sparseness of the powers p i s. Indeed, in general, weak solutions can be unbounded, as proved in [17,26]. We refer to the surveys [27,28] for an exhaustive treatment of the subject and references. The problem of regularity for anisotropic operators behaving like (1.6) with measurable and bounded coefficients remains a major challenge after more than 50 years. Recently, some progress has been made in the parabolic prototype case, as for instance in [3,31] about Lipschitz continuity, [5] about intrinsic Harnack estimates [15], and in the singular case for Barenblatt-type solutions. Moreover, as we will see, various parabolic techniques have been applied, but in no circumstance Harnack estimates have been found when more than one spatial dimension was considered. This is due to the fact that usual parabolic techniques rely on the particular structure of a first derivative in time and are not suitable to manage stronger anisotropies. With the present work, limited to the case p i = 2 for i = 1 , … , s and p i = p < 2 for i ∈ s + 1 , … , N we are able to prove a purely elliptic pointwise Harnack estimate when the operator acts on the first s variables. Furthermore, we have now a way to understand how these estimates degenerate when s varies; describing, roughly speaking, when the operator is closer to the p -Laplacian or to an uniformly elliptic operator (see the discussion after Theorem 1.2).

In the present work, we are interested in bounded solutions, therefore, leaving the problem of boundedness to the already rich literature. Our aim is to manage the anisotropic behavior of the operator interpreting its action in correspondence with a suitably adapted version of the technique developed by Chen and DiBenedetto (see the original paper [6] or the books [10,30]) in order to restore the homogeneity of the parabolic p-Laplacian. Indeed, because of the double derivative, equation (1.1) has a wilder heterogeneity of the operator than the parabolic p -Laplacian, and the intrinsic geometry will be set up according to the order and power of derivatives resulting in the dimensional analysis of the equation.

An interesting attempt in this direction has already been done by some of the authors in [23], in the case of only one nondegenerate variable (see also [25]). There an expansion of positivity is provided by applying an idea from [12], shaped on a proper exponential change of variables. Nevertheless, the change of variables in consideration is a purely parabolic tool, so that it does not allow the authors to go through more than one nondegenerate variable. The present work is conceived to fill this gap and to spread new light on the link between classical logarithmic estimates and anisotropic operators.

The two fundamental tools that we derive in our work are Theorems 1.3 and 1.4. Theorem 1.4 consists in a L 1 − L ∞ Harnack inequality, which is independent of the other two theorems, although its proof relies as well on logarithmic estimates (2.13). The name L 1 − L ∞ -inequality refers to the fact that it is possible to control the supremum of the function through some L 1 - integral norm of the function itself. This precious inequality can be used in turn to derive in straight way the Hölder continuity of solutions (see for instance [7] for a simple proof in the parabolic setting).

On the other hand, Theorem 1.3 provides both a shrinking property and an expansion of positivity. The special feature of Theorem 1.3 called shrinking property consists in the fact that from a whatever upper bound to the relative measure of some super-level of the solution it is possible to recover a pointwise estimate of positivity. Its proof is a proper consequence of logarithmic estimates (2.13) and a suitable choice of test functions (see functions f in Step 1 of the proof of Lemma 3.2). The expansion of positivity property refers to the possibility to expand along the space (in singular variables) the lower bound yet gained. The proof of this property is therefore linked to the measure theoretical approach of this shrinking property, and it is an adaptation of an idea of [13]. Here to end the proof of Theorem 1.3 we use in a crucial way the shrinking property to reach a critical mass and use Lemma 2.4.

Finally, in order to prove Theorem 1.2 we use an argument originally conceived by Krylov and Safonov in [21] to reach a certain controlled bound on the solution in terms of the solution itself, and then use repeatedly Theorem 1.4 to achieve an upper bound on the measure of some super-level set of the solution. Nonetheless, the argument of Krylov and Safonov gave us this information around an unknown point. Therefore, we apply Theorem 1.3 to expand the positivity until the desired neighborhood of the initial point and get the job done. The idea is an adaptation of the techniques originally developed in [13] to the case of anisotropic elliptic equations (1.2) and (1.3).

1.4 Structure of the article

In Section 2, we recall major functional tools and use them to derive fundamental properties of solutions as energy estimates, logarithmic estimates, and some integral estimates. Then in Section 3 we prove Theorem 1.3, in Section 4 we prove the Hölder continuity of solutions while in Section 5 we prove the L 1 − L ∞ Harnack estimate (1.17). Section 6 is devoted to the proof of Theorem 1.2. Technicalities and standard material are given in Section A, to leave space along the previous text to what is really new.

Notations:

• If Ω is a measurable subset of R N , we denote by ∣ Ω ∣ its Lebesgue measure. We will write Ω ⊂ ⊂ R N when Ω is an open bounded set.

• For r > 0 , x ¯ = ( x ¯ ′ , x ¯ ″ ) ∈ R s × R N − s , we denote by B r ( x ¯ ) the ball of radius r and center x ¯ ; the standard polydisc is denoted by Q θ , ρ = B θ ( x ¯ ′ ) × B ρ ( x ¯ ″ ) ⊂ R N . Furthermore, by w s = ∣ B 1 ( 0 ′ ) ∣ and w N − s = ∣ B 1 ( 0 ″ ) ∣ we denote the measures of the respective unit balls.

• The symbol ∀ ae stands for –for almost every–.

• For a measurable function u , by inf u and sup u we understand the essential infimum and supremum, respectively; when u : Ω → R and a ∈ R , we omit the domain when considering sub/super level sets, letting [ u ⋚ a ] = { x ∈ E : u ( x ) ⋚ a } ; if u is defined on some open set Ω ⊂ R N , we let ∂ i u = ∂ ∂ x i u denote the distributional derivatives.

• For numbers B , C > 0 we write C ∧ B = max { B , C } .

• We make the usual convention that a constant γ > 0 depending only on the data, i.e., γ = γ ( N , 2 , p , C 1 , C 2 , C ) , may vary from line to line along calculations.

2.1 Functional and standard tools

We recall the embedding W o 1 , p ( Ω ) ↪ W o 1 , p ¯ ∗ ( Ω ) proved by Troisi in [29].

It is worth pointing out that without vanishing initial datum this embedding fails in general (see [22,18] for counter-examples). A simple calculation reveals that condition p ¯ < N is always in force in our case in study. Next lemma introduces a well-known weighted Poincaré inequality (see for instance Prop. 2.1 in [10]), which will be useful when estimating the logarithmic function.

2.2 Properties of solutions to (1.2) and (1.3)

The following classical energy estimates can be proved by a standard choice of test functions.

See Section A for the classical proof. Next lemma is a sort of measure theoretical maximum principle. It asserts that if a certain sub-level set of the solution reaches a critical mass, then the solution is above a multiple of the level on half sub-level set. We agree to refer to it as usual in the literature by the epithet Critical Mass lemma (De Giorgi-type lemma is used equivalently). We state it just for sub-level sets, a similar statement being true for super-level sets.

The following lemma estimates the essential supremum of solutions by quantitative integral averages of the solution itself. Its proof is similar to the one of ([23], Prop. 8) and it is postponed to the Appendix.

Note that (2.10) with l = 1 corresponds to (1.9). Finally, we give detailed description of the main analytical tool of the present work, the following logarithmic estimates.