A priori Hölder and Lipschitz regularity for generalized $p$-harmonious functions in metric measure spaces

Abstract

Let $(\mathbb{X},d,\mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0<\varrho (x)\le \mathrm{dist}(x,\partial \Omega )$ and let $B_x$ be the closed ball centered at $x$ with radius $\varrho (x)$. If $\alpha \in \mathbb{R}$, consider the following operator in $C(\overline{\Omega })$, $${\mathcal{T}}_{\alpha }u(x)=\frac{\alpha }{2}\left(\underset{B_x}{\sup }u+\underset{B_x}{\inf }u\right)+(1-\alpha )\frac{1}{\mu (B_x)}{\int }_{B_x}ud\mu .$$ Under appropriate assumptions on $\alpha$, $\mathbb{X}$, $\mu$ and the radius function $\varrho$ we show that solutions $u\in C(\overline{\Omega })$ of the functional equation ${\mathcal{T}}_{\alpha }u=u$ satisfy a local Hölder or Lipschitz condition in $\Omega$. The motivation comes from the so called $p$-harmonious functions in euclidean domains.

22 pages

References in corpus (1)

• On the Dirichlet problem for solutions of a restricted nonlinear mean value property