Neumann problem for p-Laplace equation in metric spaces using a variational approach: existence, boundedness, and boundary regularity

TL;DR

This paper investigates the Neumann boundary value problem for the p-Laplacian in metric spaces, establishing existence, boundedness, and boundary regularity of solutions using a variational approach and advanced trace theorems.

Contribution

It introduces a variational framework for the Neumann problem in metric spaces and proves existence, boundedness, and boundary regularity of solutions, extending classical results to a more general setting.

Findings

Solutions exist and are bounded for the Neumann p-Laplacian problem.

Boundary continuity properties of solutions are established.

Utilizes trace theorem and De Giorgi inequality in the metric space context.

Abstract

We employ a variational approach to study the Neumann boundary value problem for the $p$-Laplacian on bounded smooth-enough domains in the metric setting, and show that solutions exist and are bounded. The boundary data considered are Borel measurable bounded functions. We also study boundary continuity properties of the solutions. One of the key tools utilized is the trace theorem for Newton-Sobolev functions, and another is an analog of the De Giorgi inequality adapted to the Neumann problem.