Local approximation of superharmonic and superparabolic functions in nonlinear potential theory

TL;DR

This paper demonstrates that superharmonic and superparabolic functions related to the p-Laplace and p-parabolic equations can be locally approximated by supersolutions, with applications to convergence of bounded supersolutions.

Contribution

It introduces a method to locally approximate superharmonic and superparabolic functions as limits of supersolutions with specific convergence of Riesz measures.

Findings

Superharmonic functions are locally limits of supersolutions.

Superparabolic functions can be approximated similarly in the parabolic case.

A family of bounded supersolutions has a convergent subsequence.

Abstract

We prove that arbitrary superharmonic functions and superparabolic functions related to the $p$-Laplace and the $p$-parabolic equations are locally obtained as limits of supersolutions with desired convergence properties of the corresponding Riesz measures. As an application we show that a family of uniformly bounded supersolutions to the $p$-parabolic equation contains a subsequence that converges to a supersolution.