Hardy inequalities for p-Laplacians with Robin boundary conditions

TL;DR

This paper investigates the optimal Hardy inequality constants for the p-Laplacian with Robin boundary conditions on convex domains, revealing conditions under which the best constant is explicitly determined and exploring extensions to non-convex domains.

Contribution

It establishes the exact best constant for Hardy inequalities with Robin boundary conditions on convex domains and discusses generalizations to non-convex domains.

Findings

Best constant equals ((p-1)/p)^p for convex domains with Dirichlet conditions on part of the boundary

Results extend to certain non-convex domains

Provides insights into Hardy inequalities with Robin boundary conditions

Abstract

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.