# Geometric regularity for elliptic equations in double-divergence form

## TL;DR

This paper investigates the regularity of solutions to double-divergence elliptic equations, showing improved continuity properties near zero level-sets, with results depending on the regularity of coefficients.

## Contribution

It establishes new regularity results for solutions based on the regularity of coefficients, using geometric and approximation techniques.

## Key findings

- Solutions are locally $ ext{C}^{1^-}$ with $ ext{H"older}$ coefficients.
- Solutions are $ ext{C}^{1,1^-}$ with Sobolev differentiable coefficients.
- Enhanced regularity along nonphysical free boundaries.

## Abstract

In this paper, we examine the regularity of the solutions to the double-divergence equation. We establish improved H\"older continuity as solutions approach their zero level-sets. In fact, we prove that $\alpha$-H\"older 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.