Variational Estimates for Nonnegative Harmonic Functions

Jakob Fromherz

Variational Estimates for Nonnegative Harmonic Functions

Analysis of PDEs 2026-07-29 v1

Authors: Jakob Fromherz

Abstract

Consider a nonnegative harmonic function $u$ on a domain $D\subset\mathbb{R}^{d-1}$ . We establish boundedness of Jones' variational integral on an ultradense, $d-1$ dimensional subset of the boundary of $C^{1,\mathrm{Dini}}$ domains. This result was previously only known for the unit ball.

Cite

@article{arxiv.2607.26663,
  title  = {Variational Estimates for Nonnegative Harmonic Functions},
  author = {Jakob Fromherz},
  journal= {arXiv preprint arXiv:2607.26663},
  year   = {2026}
}