Neumann $p$-Laplacian problems with a reaction term on metric spaces

TL;DR

This paper investigates the existence and regularity of solutions to Neumann $p$-Laplacian problems with reaction terms on metric spaces using variational methods, trace theorems, and De Giorgi conditions.

Contribution

It introduces a variational framework for Neumann $p$-Laplacian problems on metric spaces with new regularity results and applies trace theorems for BV functions.

Findings

Existence of solutions established under specified conditions.

Regularity results for solutions derived using De Giorgi type conditions.

Application of trace theorems for BV functions in the variational setting.

Abstract

We use a variational approach to study existence and regularity of solutions for a Neumann $p$-Laplacian problem with a reaction term on metric spaces equipped with a doubling measure and supporting a Poincar\'e inequality. Trace theorems for functions with bounded variation are applied in the definition of the variational functional and minimizers are shown to satisfy De Giorgi type conditions.