Uniqueness for local-nonlocal elliptic equations

TL;DR

This paper establishes the conditions for uniqueness of solutions to mixed local-nonlocal elliptic equations with variable coefficients, highlighting the optimality of these conditions and exploring the related parabolic equations.

Contribution

It provides the first comprehensive analysis of uniqueness for mixed local-nonlocal elliptic equations with variable coefficients, including optimal conditions and their implications.

Findings

Uniqueness holds under specific assumptions on the coefficient behavior at infinity.

Failure of these assumptions leads to nonuniqueness of solutions.

The results extend to the parabolic counterpart of the equations.

Abstract

We study mixed local and nonlocal elliptic equation with a variable coefficient $\rho$. Under suitable assumptions on the behaviour at infinity of $\rho$, we obtain uniqueness of solutions belonging to certain weighted Lebsgue spaces, with a weight depending on the coefficient $\rho$. The hypothesis on $\rho$ is optimal; indeed, when it fails we get nonuniqueness of solutions. We also investigate the parabolic counterpart of such equation.