The double phase Dirichlet problem when the lowest exponent is equal to 1

TL;DR

This paper establishes existence and uniqueness for the double phase Dirichlet problem at the critical exponent 1, using BV functions and generalized Orlicz-Sobolev spaces, advancing understanding of such PDEs.

Contribution

It introduces a novel approach to the double phase Dirichlet problem at exponent 1, employing BV functions and limits of solutions with varying exponents.

Findings

Existence and uniqueness of solutions at exponent 1

Solutions are functions of bounded variation

Development of properties of generalized Orlicz-Sobolev spaces

Abstract

In this paper we prove an existence and uniqueness result for the double phase Dirichlet problem when the lowest exponent is equal to 1. Our solution is a function of bounded variation that simultaneously lies in a suitable weighted Sobolev space and is found as the limit of a sequence of solutions of intermediate double phase Dirichlet problems whose lowest exponent $p$ goes to 1. As a result of that, our approach involves the study of some relevant properties of generalized Orlicz-Sobolev spaces.