Vol. 13, No. 4, 2020

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Geometric regularity for elliptic equations in double-divergence form

Raimundo Leitão, Edgard A. Pimentel and Makson S. Santos

Vol. 13 (2020), No. 4, 1129–1144
Abstract

We examine the regularity of the solutions to the double-divergence equation. We establish improved Hölder continuity as solutions approach their zero level-sets. In fact, we prove that α-Hölder continuous coefficients lead to solutions of class 𝒞1 , locally. Under the assumption of Sobolev-differentiable coefficients, we establish regularity in the class 𝒞1,1 . Our results unveil improved continuity along a nonphysical free boundary, where the weak formulation of the problem vanishes. We argue through a geometric set of techniques, implemented by approximation methods. Such methods connect our problem of interest with a target profile. An iteration procedure imports information from this limiting configuration to the solutions of the double-divergence equation.

Keywords
double-divergence equations, geometric regularity, improved regularity at zero level-sets
Mathematical Subject Classification 2010
Primary: 35B65, 35J15
Milestones
Received: 8 August 2018
Revised: 18 March 2019
Accepted: 18 April 2019
Published: 13 June 2020
Authors
Raimundo Leitão
Department of Mathematics
Universidade Federal do Ceara
Fortaleza
Brazil
Edgard A. Pimentel
Department of Mathematics
Pontifical Catholic University of Rio de Janeiro
Rio de Janeiro
Brazil
Makson S. Santos
Department of Mathematics
Pontifical Catholic University of Rio de Janeiro
Rio de Janeiro
Brazil