Variational methods for nonpositive mixed local-nonlocal operators

TL;DR

This paper establishes the existence of weak solutions for boundary value problems involving mixed local-nonlocal operators, even when these operators are not positive definite, expanding the scope of variational methods.

Contribution

It introduces a novel approach to handle nonpositive definite mixed local-nonlocal operators in boundary value problems.

Findings

Existence of weak solutions proven for nonpositive mixed operators

Extension of variational methods to nonpositive definite cases

Applicable to a broader class of boundary value problems

Abstract

We prove the existence of a weak solution for boundary value problems driven by a mixed local--nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite.