A PDE approach to nonlinear potential theory in metric measure spaces

TL;DR

This paper develops a PDE-based framework for nonlinear potential theory in metric measure spaces, providing new insights into p-harmonic functions and their properties in this setting.

Contribution

It introduces a PDE description of p-harmonic functions in metric measure spaces and applies it to establish new results on harmonic functions and Busemann functions.

Findings

Sheaf property of harmonic functions established

PDE proof that convex composition preserves subminimizers

Busemann functions are harmonic in infinitesimally Hilbertian CD(0,N) spaces

Abstract

We show that the tools recently introduced by the first author in [9] allow to give a PDE description of p-harmonic functions in metric measure setting. Three applications are given: the first is about new results on the sheaf property of harmonic functions, the second is a PDE proof of the fact that the composition of a subminimizer with a convex and non-decreasing function is again a subminimizer, and the third is the fact that the Busemann function associated to a line is harmonic on infinitesimally Hilbertian CD(0,N) spaces.