On the Local Behavior of Local Weak Solutions to some Singular Anisotropic Elliptic Equations

We study the local behavior of bounded local weak solutions to a class of anisotropic singular equations that involves both non-degenerate and singular operators. Throughout a parabolic approach to expansion of positivity we obtain the interior H\"older continuity, and some integral and pointwise Harnack inequalities.


Introduction
In this note we study local regularity properties for bounded weak local solutions to operators whose prototype is having a non-degenerate behavior along the first s-variables, and a singular behavior on the last ones. This kind of operators are useful to describe the steady states of non-Newtonian fluids that have different directional diffusions (see for instance [1]), besides their pure mathematical interest, which still is a challenge after more than fifty years. Precise hypothesis will be given later (in Section 1.2), leaving here the space to describe what are the novelties and significance of the present work in the context of this kind of operators.
Until this moment it is not known whether solutions to equations as (1.1) enjoy the usual local properties as p-Laplacean ones. This is because equation (1.1) is part of a more general group of operators, whose regularity theory is still fragmented and largely incomplete. It is clear that new techniques are needed for a correct interpretation of the problem and its resolution. The present work is conceived to introduce a new method, adapted from the theory of singular parabolic equations. In next Section we explain this simple but effective idea, that we will apply to a class of equations as (1.1), that have no homogeneity on the differential operator (hence the epithet anisotropic) because they combine both nondegenerate and singular properties.

The 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 2r (x o ) be the ball of radius 2r and center x o . Now for 0 < t < r such that B 2r (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 dH 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 get 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.

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 ∂ i A i (x, u, ∇u) = 0, weakly in Ω, (1.2) 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, |A i (x, u, ξ)| ≤ C 2 |ξ i | p−1 + C, for i ∈ {s + 1, .., N }, 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 All along the present work 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 then by a simple limit argument it can be shown that (1.5) is always in force (see for example [10], Lemma 1.1 Chap. II).
Properties of anisotropic Sobolev spaces have first been investigated in [22], [29], [18], 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 literature that local weak solutions to our 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.
We state our two main results hereafter. The first one is a result of local Hölder continuity. Then there exists α ∈ (0, 1) depending only on the data such that u ∈ C 0,α loc (Ω).
Next we fix some geometrical notations and conventions. For a point x o ∈ Ω, let us denote it by 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.
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). . Fix a pointx ∈ Ω and numbers θ, ρ > 0 such that Q 8θ,8ρ (x) ⊂ Ω. Then there exists a positive constant γ depending only on the data such that either (1.16) If additionally property (1.9) holds, then either we have (1.15) or (1.17)

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 mayor challenge after more than fifty years. Recently some progresses have been made in the parabolic prototype case, as for instance in [3], [31] about Lipschitz continuity, [5] about intrinsic Harnack estimates and [15] 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 adaptd version of the technique developed by E. 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 the 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, 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)-(1.3).

Structure of the paper
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 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 has been collected in a final section, Section 7, 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.
the ball of radius r and center x; the standard polydisc is denoted by 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 = ∂ ∂xi 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.

Preliminaries
In this Section we collect the basic tools that will be used along the overall theory. For the sake of readability, simpler and well-known proofs are postponed to the Appendix (Section 7), while most relevant passages that bring to light our method are detailed and highlighted.

Functional and standard tools
We recall the embedding (Ω) proved by M. Troisi in [29].
Then there exists a positive constant γ(N,p) such that 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 conditionp < 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]), that will be useful when estimating the logarithmic function.
Lemma 2.2. Let B ρ be a ball of radius ρ > 0 about the origin, and let ϕ ∈ C(B ρ ) satisfy 0 ≤ ϕ(x) ≤ 1 for each x ∈ B ρ together with the condition that the level sets [ϕ > k] ∩ B ρ are convex for each k ∈ (0, 1). Let g ∈ W 1,p (B ρ ), and assume that the set has positive measure. The there exists a constant γ > 0 depending only upon N, p such that

Properties of solutions to (1.2)-(1.3)
The following classical Energy Estimates can be proved by a standard choice of test functions.
(2.4) See Section 7 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 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.
Then there exists a number ν ∈ (0, 1) whose dependence from the data is specified by (2.9) and such that if Proof. We suppose without loss of generality thatx = 0. For j = 0, 1, 2.. let us set and let ζ j ∈ C ∞ o (Q j ) be a cut-off function between Q j and Q j+1 such that ζ j ≡ 1 on Q j+1 ,0 ≤ ζ j ≤ 1 and therefore satisfying Then, combining a precise use of Hölder inequality and Troisi's embedding (2.2) to the energy estimates (2.4), leads us to the estimate (2.8) By assumption ξω > ρ the third term on right hand side is smaller than 1. If we define Y j = |A j |/|Q j |, we divide (2.8) by |Q j+1 | and we observe that |Q j | ≤ γ2 j |Q j+1 | ≈ (θ s ρ N −s ), previous estimate can be written as by simple manipulation on the various exponents. We evoke Lemma 7.1 to declare that if sets ν free from any other dependence than the initial data.
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. (2.10) Then there exist constants γ, C > 1 depending only on the data, such that for all polydiscs Q 2θ,2ρ ⊂ Ω we have either (2.12) 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.

(2.13)
Proof. We test the equation (1.5) with a nonnegative function ) a test function, and we use the structure conditions (1. 3) to get the integrals being taken over the polydisc Q θ,ρ (x). We use repeatedly Young's inequality to get, for the first term on the right while the third term on the right is estimated with Similarly fourth term is estimated by Gathering all the pieces together and choosing accordingly ǫ small enough to reabsorb on the left the energy terms, we get Finally, choose 0 ≤ ζ(x ′′ ) ≤ 1 and φ(x ′ ) = (t 2 − |x ′ −x ′ | 2 ) + and estimate the second term on the right by use of Green-Ostrogradsky's formula with (2.14) and the proof is concluded.

Proof of Theorem 1.3
We start by proving two main Lemmas, whose combination will provide an easy proof of Theorem 1.3. The first one turns a measure estimate given on the intrinsic polydisc into a measure estimate on each (N − s)-dimensional slice. Assume that for some M, ρ > 0 and some α ∈ (0, 1) the following estimate holds

1)
for some δ < α 4 (1 + α) s /(1 + α 2 ). Then there exist numbers s 1 > 1, δ 1 ∈ (0, 1) depending only on the data and α such that Proof. Let σ, δ 1 ∈ (0, 1) to be chosen later, let ζ( Let y ′ ∈ B (1−δ1)θ (x ′ ) be a Lebesgue point for the function let us call z = (y ′ ,x ′′ ) and use just the right hand side of inequality (2.13), with f (u) = u, Now we integrate this inequality on t ∈ (0, δ 1 θ) and estimate the various terms. The first term can be evaluated by using that y ′ is a Lebesgue point. Second term is estimated with and third term similarly. Gathering all together we obtain We observe that B δ1 (y ′ ) ⊂ B θ (x ′ ) and that by imposing the natural intrinsic geometry the term in parenthesis can be ruled. Indeed we let θ = ρ p/2 (δN ) (2−p)/2 and compute using M ≥ ρ and the hypothesis (3.1). We estimate from below in the whole B ρ (x ′′ ) by Combining this remark with previous calculations we have the inequality To conclude the proof we choose δ 1 ,s 1 ,σ,δ, from the conditions Second and following Lemma is what is called in literature a shrinking Lemma. Indeed, from a given relative measure information on a level set, it allows to shrink as much as we need the relative measure on a lower-level set. Even more interesting, it provides also an expansion of positivity along singular variables.

By transformation Φ inequality (3.4) turns into
The expansion of positivity relies on the following simple fact. The inequality above implies the measure estimate for ) be a convex cut-off function between balls B 1 and B 1/2 , i.e.
Let us fix numbers j * ∈ N and ǫ ∈ (0, 1) and for j = 1, 2, .., j * we define Then, inequality (2.13) reads, for t ∈ (0, 1/2), (3.11) Last two terms on the right of (3.11) can be reduced, by assuming M ǫ j * > ρ, to We observe that first integrands on the left of (3.11) are the directional derivatives of and, as g ∈ W 1,p (B 1 (0 ′′ )) vanishes in [v > e j ] ∩ B 1 |, we can apply the weighted Poincaré inequality (2.3), using (3.10) to estimate the term |E| A precise analysis of this last simple fact reveals its correspondence with (3.9)-(3.10) in terms of expansion of positivity. Putting all the pieces of the puzzle into (3.11) we arrive finally to the following logaritmic estimate, valid for each j = 1, .., j * , and 0 < t < θ, Let us define A j (t) = [v < ǫ j ] ∩ Q t,1 (0) and let Now we show that if M ≥ Kρ, then there exists a number ξ = ξ(ν) ∈ (0, 1) such that y j+1 ≤ max{ν, (1 − ξ)y j } for each j = 1, .., j * . This, with a standard iteration procedure will end the proof. So we proceed by assuming y j+1 > ν, and by continuity of the integral we can choose t o ∈ (0, 1/2) such that Y j+1 (t o ) = y j+1 and divide the argument in two alternatives. Let and suppose Ψ ′ (t o ) ≤ 0. Then inequality (3.12) implies that for each fixed σ ∈ (0, 1) giving the estimate Now we determine σ, ǫ small enough to get y j+1 ≤ ν. Indeed, by |Q t,1 | ≤ γt −s |B 1 (0 ′′ )| we see that the following estimate holds Let us suppose now that Ψ ′ (t o ) > 0 and that there exists t * = inf{t ∈ (t o , 1/2)| Ψ ′ (t) ≤ 0}, so that by definition Ψ is monotone increasing before t * and we have For the time t * similar estimates to (3.12), (3.15), (3.17) hold and we obtain similarly that In this case the value of ǫ has been already chosen, so that last inequality is valid provided we restrict to levels τ such that for a function h(ν) = o(ν 3/p ). Finally, we use (3.20) and (3.19) together with Fubini's theorem and a change of variables to estimate where we used y j > Y j+1 (t * ) in first inequality and y j > y j+1 > ν in last one. By an easy manipulation we see that we can bound from below (3.19), indeed (3.23) So we gather (3.19), (3.22) and (3.23) together to get

(3.24)
Finally, this implies redefining ǫ = min{1/2, ǫ} if needed. We prove now that if t * does not exist, then the iteration inequality above is still satisfied. Indeed in case no such t * exists then Ψ ′ (t) > 0 for all t ∈ [t o , 1/2] and therefore Ψ(t o ) ≤ Ψ(1/2). Moreover by simple calculations analogous to (3.23), (3.24) we recover the estimates + and use Young inequality we can derive, similarly to (3.12), the estimate From this, by (3.26) we obtain and therefore estimating (3.27) from above we conclude that y j+1 ≤ ν by choosing ǫ small enough. Finally, for the sake of readability we just remark that in case Ψ is not regular enough it is possible to perform the same argument above by substituting Ψ ′ with its right Dini derivative, as in ( [6], Sec. 7).

Conclusion of the Proof of Theorem 1.3
Letx ∈ Ω, ρ > 0 and α ∈ (0, 1). We suppose that for δ(α) ∈ (0, 1) and M > 0 we have the information By Lemma 3.1 there exist numbers s 1 , δ 1 > 0 depending only from the data and α such that either M ≤ ρ or For ν as in (2.9), we apply Lemma 3.2 with so that there exist numbers K > 1, δ o ∈ (0, 1) depending on the data and α, ν such that either M ≤ Kρ or To recover the correct intrinsic geometry (see Remark 2.1), we cut the slice-wise information on polydisc Qθ ,4ρ (x) to an information on a polydisc which is smaller along the nondegenerate variables, by so that θ o ≤θ, increasing j * in (3.28) in case of need. Finally we apply the Critical Mass Lemma 2.4 to end the proof.

Hölder Continuity. Proof of Theorem 1.1
We begin with the accommodation of degeneracy. Let x 0 ∈ Ω be an arbitrary point, M = sup Ω |u| and ρ > 0 such that Once we have this kind of controlled reduction of oscillation the whole procedure can be iterated in nested shrinking polydiscs and the rest of the proof is standard (see for instance Theorem 3.1 in [11], Chap. X).
5 L 1 − L ∞ estimates. Proof of Theorem 1.4 Let us fix a pointx ∈ Ω and numbers ρ, θ > 0 such that Q 8θ,8ρ (x) ⊂ Ω. Let y ′ ∈ B θ/2 (x ′ ) be a Lebesgue point for the function For σ ∈ (0, 1) we consider a generic radius r such that be a cut off function relative to the last N − s variables between B (1−σ)r and B r satisfying We divide the proof in three steps. For ease of notation, let us call Q t,r := Q t,r (y ′ ,x ′′ ) and (5.1) Now we estimate (5.1) by use of structure conditions (1.3) to get using Hölder inequality and multiplying and dividing for (u + η) β(p−1)/p , β > 0.
Let us estimate the second integral term in (5.2). Let ζ(x ′′ ) be as before and let us test the Using structure conditions (1.3) and Young's inequality, we get Qt,r Applying repeatedly Young's inequality and reabsorbing on the left the terms involving energy estimates we obtain We estimate separately the various terms. For the first one we have Next we use 1 < p < 2 to split (u + η) p−β = (u + η) p−2 (u + η) 2−β ≤ η p−2 (u + η) 2−β to get Inequalities above about I 1 , I 2 and (5.4) lead us to the formula N i=s+1 Qt,r We put (5.5) inside (5.2), summing (N − s) times the same quantity, to obtain Now we evaluate by Jensen's inequality separately for α = 2 − β, β(p − 1), the terms and therefore we estimate from above inequality (5.6) by Now we divide left and right-hand side of this inequality for t s+1 ρ N −s , we take the supremum on times 0 < t < 2θ on the right and we integrate between 0 ≤ τ ≤ t, to get Finally we use that y ′ is a Lebesgue point and that t < 2θ to have the estimate (5.7)

STEP 2. INTEGRAL INEQUALITY (5.7) IMPLIES A NONLINEAR ITERATION
We consider ε ∈ (0, 1) and use Young's inequality to (5.7) to get for the first term Then we require a condition β < p (p−1) to get for the second term Finally with these specifications, formula (5.7) is majorized by beingγ > 0 a constant depending only on the data. The term in parenthesis is smaller than one if we contradict condition (1.15), and the right-hand side of (5.8) is estimated from above with Since t ∈ (0, 2θ) is an arbitrary number we choose the timet that achieves the supremum on the left of previous formula, and we observe that the right hand side of (5.9) does not depend on t. This implies We consider the increasing sequence ρ n = (1 − σ n )r = ρ( n i=1 2 −i ) → 2ρ and the expanding polydisc Q n = {Q t,ρn } : Q t,ρ → Q t,2ρ . By generality of the choice of r, previous formula implies the recurrence with b = 2γ. Therefore with an iteration as in (7.6) we arrive to the conclusion We consider formula (2.12) with l = 1, which is (5.12) Let us insert in the integral term on the right of (5.12) our previously obtained formula (5.11) to get The proof is complete. 6 Harnack inequality. Proof of Theorem 1.2 Here we prove the Harnack estimate (1.11), by use of Theorems 1.3, 1.4. Without loss of generality we assume that x 0 = 0 and denote u(x 0 ) = u 0 to ease notation. First we begin with an estimate reminescent of [21] aiming to a bound from above and below in terms of the radius itself (estimate (6.2)).  For a parameter λ ∈ (0, 1) we consider the equation Let λ 0 be the maximal root of the equation (6.1) and by continuity let us fix a pointx ′′ by beingx ′′ a point in B λ0ρ (0 ′′ ). See Figure 1 for a drawing. Now let us define λ 1 ∈ (0, 1) by and set also 2r : Then by definition of λ 0 it holds both sup See Figure 1 for this construction. Henceforth we arrive to the estimate Now we use the bound obtained joint to the L 1 − L ∞ estimate (1.17) to reach an estimate of the measure of sub-level sets of u, in order to apply Theorem 1.3. We consider the measure estimate (6.6): either holds (1.10) or This implies the measure estimate We can apply again Theorem 1.3. This time and next ones being α = 1/2 fixed, there exists a number δ * depending only on the data and such that for almost every x ∈ Q η * ,2r (0 ′ ,x ′′ ) we have Now the procedure can be iterated a number n ∈ N of times such that n ≥ log 2 (4/(1 − λ 0 )) in order to have We observe that in previous calculation of η * (n) the powers of 2 cancel each other out.
. If we assume β > 2 the choice ofn above implies 2nr ≥ 2ρ so that by properly choosing β ≥ p/(2 − p) and redefining the constants. Observe that equation (6.9) is exactly (1.11). To end the proof, we will choose β big enough to free the lower bound u(x) > (δ * /2) n M * , by any dependence of the solution itself other than u 0 . Indeed, decreasing ρ in case of need, let n ∈ N be a number big enough that n ≥n, Then we have Decreasing δ * in case of need, we choose finally β > p/(2 − p) so big that and the claim follows.

Appendix
In this Appendix we enclose all the details that for reader's convenience have been postponed.
Gathering together all this estimates and choosing ǫ > 0 to be a constant small enough concludes the proof.

Iteration Lemmata
Here we recall two basic Lemmas, extremely useful for the iteration techniques employed in our analysis and whose proofs can be found in ( [10], Chap. I Sec.IV). where C, b > 1 and α ∈ (0, 1) are given numbers. Then we have the logical implication The following Lemma is useful for reverse recursive inequalities of Section 7.3.
With these stipulations,the energy estimates (2.4) are being A n = Q n ∩ [u > k n+1 ]. Now for any s > 0 we observe that This fact with s = 2 together with Hölder inequality gives Putting this into the energy estimates above leads us to because k ≥ (θ 2 /ρ p ) 1 2−p ≥ ρ. Now an application of Troisi's Lemma 2.1 and (7.8) above give us the following inequality