On higher integrability estimates for elliptic equations with singular coefficients

2018/04/09 by Juraj Földes, Földes, Juraj, Tuoc Phan +1
Engineering · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1804.03180

openalex publication_date 2018/04/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

In this note we establish existence and uniqueness of weak solutions of linear elliptic equation div[A(x) ∇ u] = divF(x), where the matrix A is just measurable and its skew-symmetric part can be unbounded. Global reverse Hölder's regularity estimates for gradients of weak solutions are also obtained. Most importantly, we show, by providing an example, that boundedness and ellipticity of A is not sufficient for higher integrability estimates even when the symmetric part of A is the identity matrix. In addition, the example also shows the necessity of the dependence of α in the Hölder C^α-regularity theory on the \textupBMO-semi norm of the skew-symmetric part of A. The paper is an extension of classical results obtained by N. G. Meyers (1963) in which the skew-symmetric part of A is assumed to be zero.

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