Geometric regularity for elliptic equations in double-divergence form

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 ${\mathcal{𝒞}}^{1^{-}}$, locally. Under the assumption of Sobolev-differentiable coefficients, we establish regularity in the class ${\mathcal{𝒞}}^{1,1^{-}}$. 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.