Optimal Hardy-weights for elliptic operators with mixed boundary conditions

TL;DR

This paper develops a method to construct optimal Hardy-weights for certain elliptic operators with mixed boundary conditions, advancing the understanding of their criticality properties.

Contribution

It introduces a new approach to generate optimal Hardy-weights for subcritical elliptic operators with mixed boundary conditions, based on recent criticality theory.

Findings

Constructed families of optimal Hardy-weights for elliptic operators.

Established conditions under which the operators become critical and null-critical.

Extended criticality theory to operators with degenerate mixed boundary conditions.

Abstract

We construct families of optimal Hardy-weights for a subcritical linear second-order elliptic operator $(P,B)$ with degenerate mixed boundary conditions. By an optimal Hardy-weight for a subcritical operator we mean a nonzero nonnegative weight function $W$ such that $(P-W,B)$ is critical and null-critical with respect to $W$. Our results rely on a recently developed criticality theory for positive solutions of the corresponding mixed boundary value problem.