Sharp trace and Korn inequalities for differential operators

TL;DR

This paper develops sharp boundary trace and Korn inequalities for vectorial differential operators, including challenging cases like p=1, using novel approaches that do not rely on global singular integral estimates.

Contribution

It introduces a new method for sharp Besov boundary traces via Riesz potentials, classifies boundary trace and Korn inequalities, and extends results to John domains without relying on Calderón-Zygmund theory.

Findings

Sharp trace estimates for p=1 via Riesz potentials

Korn inequalities hold on John domains for 1<p<∞

Reduction of trace estimates to gradient estimates despite lack of Calderón-Zygmund theory

Abstract

We establish sharp trace- and Korn-type inequalities that involve vectorial differential operators, the focus being on situations where global singular integral estimates are not available. Starting from a novel approach to sharp Besov boundary traces by Riesz potentials and oscillations that equally applies to $p=1$, a case difficult to be handled by harmonic analysis techniques, we then classify boundary trace- and Korn-type inequalities. For $p=1$ and so despite the failure of the Calder\'{o}n-Zygmund theory, we prove that sharp trace estimates can be systematically reduced to full $k$-th order gradient estimates. Moreover, for $1<p<\infty$, where sharp trace- yield Korn-type inequalities on smooth domains, we show for the basically optimal class of John domains that Korn-type inequalities persist -- even though the reduction to global Calder\'{o}n-Zygmund estimates by extension operators might not be possible.