Partial regularity of the gradient for subsolutions

Aram Hakobyan; Michael Poghosyan; Henrik Shahgholian

Partial regularity of the gradient for subsolutions

Analysis of PDEs 2026-02-18 v2

Authors: Aram Hakobyan , Michael Poghosyan , Henrik Shahgholian

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.

Keywords

harmonic functions dirichlet problem pseudoconvex domains

Cite

@article{arxiv.2602.14305,
  title  = {Partial regularity of the gradient for subsolutions},
  author = {Aram Hakobyan and Michael Poghosyan and Henrik Shahgholian},
  journal= {arXiv preprint arXiv:2602.14305},
  year   = {2026}
}