Coupling local and nonlocal equations with Neumann boundary conditions

TL;DR

This paper presents two novel methods for coupling local and nonlocal equations with Neumann boundary conditions, ensuring the models are linked to an energy functional and have unique minimizers up to a constant.

Contribution

It introduces two new coupling approaches for local and nonlocal equations with Neumann conditions, establishing existence and uniqueness of energy minimizers.

Findings

Existence of energy minimizers for the coupled models

Uniqueness of minimizers modulo constants

Two different coupling methods proposed

Abstract

We introduce two different ways of coupling local and nonlocal equations with Neumann boundary conditions in such a way that the resulting model is naturally associated with an energy functional. For these two models we prove that there is a minimizer of the resulting energy that is unique modulo adding a constant.