Variational Estimates for Nonnegative Harmonic Functions

Jakob Fromherz

partial differential equations

Variational Estimates for Nonnegative Harmonic Functions

arXiv:2607.26663

summary

The paper proves that Jones' variational integral is bounded for nonnegative harmonic functions on an ultradense subset of the boundary of $C^{1, e x t D ini}$ domains, extending a result previously known only for the unit ball.

Abstract

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

Topics & keywords

#harmonic functions #variational integrals #boundary regularity #c1,dini domains #jones integralnonnegative harmonic functionJones' variational integralultradense subsetC^{1,Dini} domainboundedness