Sobolev trace inequalities on John domains and its applications

TL;DR

This paper establishes Sobolev trace inequalities on a broad class of John domains, including those near convex bodies, without requiring boundary regularity, and applies these results to the quantitative Wulff inequality.

Contribution

It proves trace inequalities on John domains with minimal boundary regularity assumptions and offers an alternative proof for a key step in the quantitative Wulff inequality.

Findings

Trace inequality holds for John domains with measure-theoretic boundary conditions.

Includes domains close to convex bodies, even without Ahlfors regularity.

Provides an alternative proof for a step in the quantitative Wulff inequality.

Abstract

We prove that a trace inequality holds for John domains $\Omega$ satisfying $$ \mathcal H^{n-1}(\partial \Omega\setminus \partial_*\Omega)=0,$$ where $\partial_*\Omega$ denotes the measure-theoretic boundary, together with an upper density bound on $\partial \Omega$. This class of domains includes $(\epsilon,\,r)$-perimeter minimizers of Wulff perimeter $P_K$ which are close to the associated convex body $K$. Particularly, this result is established without requiring $\partial \Omega$ to be Ahlfors regular. As a consequence, we give an alternative proof for a crucial step in the quantitative Wulff inequality, thereby providing a meaningful commentary on the seminal work of Figalli, Maggi, and Pratelli \cite{FMP2010}.