Regularity for quasilinear equations on degenerate singular sets

Abstract

We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy S(u) := {X : Du(X) = 0}. Our main Theorem reveals that along S(u), u is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set S(u), Du enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. The results are new even in the context of linear elliptic equations, where it is herein shown that H 1 -weak solutions to div (aij (X)Du) = 0, with aij elliptic and Dini-continuous are actually C 1,1 − along S(u). The results and insights of this work foster a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.

Classification of positive D1,p(RN)-solutions to the critical p

“…Therefore, for a, A fixed, it follows that w Rn is uniformly bounded in L ∞ (B A \ B a ) and hence, by [9,13,14,21,23], it is also uniformly bounded in C 1,α (K), 0 < α < 1, for any compact set K ⊂ B A \ B a . We agree that a, A are redefined so that the C 1,α estimates holds in the closure of B A \ B a .…”

Regularity and comparison principles for p-Laplace equations with vanishing source term

“…We start recalling that, since we assumed that (1.3) is satisfied, then the results in [27] apply and (1.4) holds. Let us fix some notations.…”

“…It is in any case important to state explicitly (1.3) since the parameter s will appear in our statements and in some cases it could be different by the one obtained by embedding theorems. Let us also mention that in [27] more general problems are considered including in particular operators with variable coefficients.…”

“…We will show here that in general it is not necessary. One of the purposes of this paper is to point out the strong relation between the study of the integrability properties of the second derivatives of the solutions and the study of the maximal exponent for the Hölder continuity of the gradient, starting from a recent important result of Teixeira [27] which is based on previous estimates obtained in [11]. Namely, exploiting the results in [11] and also some techniques from [1], it has been shown in [27] that, if u ∈ W 1,p (Ω) is a solution to (1.1) and f ∈ L s (Ω) with s > n, (1.3) then…”

Regularity for the fully nonlinear dead-core problem

“…therefore the estimate provided in Theorem 2 is superior than the one coming from Schauder regularity theory-compare with [11,12]. An important application of the universal estimate provided by Theorem 2 is delivered in Theorem 8, where a sharp Liouville type of theorem is proven for entire solutions to (1.2).…”