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
-Hölder
continuous coefficients lead to solutions of class
, locally.
Under the assumption of Sobolev-differentiable coefficients, we establish regularity in the
class
.
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