#### 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 $\alpha$-Hölder continuous coefficients lead to solutions of class ${\mathsc{𝒞}}^{{1}^{-}}$, locally. Under the assumption of Sobolev-differentiable coefficients, we establish regularity in the class ${\mathsc{𝒞}}^{1,{1}^{-}}$. 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