Extremal functions for Morrey's inequality in convex domains

TL;DR

This paper investigates extremal functions related to Morrey's inequality in convex domains, showing their ratios are constant, and explores conditions under which this property holds or fails, also linking to Green's function uniqueness.

Contribution

It demonstrates that in convex domains, extremal functions for Morrey's inequality have constant ratios, and clarifies the role of convexity in this property, including non-convex cases.

Findings

Ratios of extremal functions are constant in convex domains.

Convexity is sufficient but not necessary for the property.

Links to uniqueness of Green's function optimization problems.

Abstract

For a bounded domain $\Omega\subset \mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\|u\|^p_{\infty}\le \int_\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\Omega)$. We show that the ratio of any two extremal functions is constant provided that $\Omega$ is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the $p$-Laplacian.