Sharp regularity for general Poisson equations with borderline sources

2011/09/22 by Eduardo V. Teixeira, Teixeira, Eduardo V.
Mathematics · #35B65 #35J70 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35B65 #msc:35J70

paper · pdf · doi:10.48550/arxiv.1109.4768

Review from previous version. Accepted for Publication: Journal de Mathématiques Pures et Appliquées

arxiv created 2012/04/26 · arxiv updated 2012/04/27

Abstract

This article concerns optimal estimates for non-homogeneous degenerate elliptic equation with source functions in borderline spaces of integrability. We deliver sharp Hölder continuity estimates for solutions to p-degenerate elliptic equations in rough media with sources in the weak Lebesgue space L_weak^{(n)/(p) + ε}. For the borderline case, f ∈ L_weak^(n)/(p), solutions may not be bounded; nevertheless we show that solutions have bounded mean oscillation, in particular John-Nirenberg's exponential integrability estimates can be employed. All the results presented in this paper are optimal. Our approach is based on powerful Caffarelli-type compactness methods and it can be employed in a number order situations.

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