Partial regularity of the gradient for subsolutions

TL;DR

This paper establishes the upper semi-continuity of the gradient for bounded subharmonic functions under specific geometric conditions on super-level sets, extending results to general operators and boundary behaviors.

Contribution

It introduces a new geometric condition involving uniform $C^{1, ext{Dini}}$ domains that ensures gradient regularity for subsolutions and solutions of elliptic equations.

Findings

Gradient of bounded subharmonic functions is upper semi-continuous under the geometric condition.

Extension of results to a class of general operators.

Application to boundary behavior of solutions in domains with $C^{1, ext{Dini}}$ boundaries.

Abstract

We prove that the gradient of any bounded subharmonic function is upper semi-continuous, provided that its super-level sets can be touched from the exterior by uniform $C^{1,\text{Dini}}$ domains at every point. This idea extends to a class of general operators, as well as to the boundary behaviour of the gradient of solutions of the Dirichlet problem in a domain whose boundary satisfy this geometric condition.