A double-phase Neumann problem with $p=1$

TL;DR

This paper investigates a double-phase Neumann problem with boundary conditions where the lower exponent is 1, establishing existence of solutions as limits of problems with higher exponents and providing a variational characterization.

Contribution

It introduces a novel approach to analyze Neumann problems with p=1 by connecting solutions to those with p>1 and offers a variational framework for the limit case.

Findings

Existence of solutions for p=1 established as limits of p>1 problems

Variational characterization of the p=1 solution provided

Method extends understanding of double-phase problems with non-homogeneous boundary conditions

Abstract

We study a double-phase Neumann problem with non-homogeneous boundary conditions, where the lowest exponent $p$ is equal to 1. The existence of a solution is established as the limit of solutions to corresponding double-phase problems with $p>1$. We also provide a variational characterization of the limit.