Partial regularity of the gradient for subsolutions

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.