Sobolev trace inequalities on John domains and its applications

Abstract

We prove that a trace inequality holds for John domains $\Omega$ satisfying $${\mathcal{H}}^{n-1}(\partial \Omega \setminus {\partial }_{\ast }\Omega )=0,$$ where ${\partial }_{\ast }\Omega$ 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}.

arXiv comment22 Pages. We are particularly grateful for the insightful observation that the result concerning the John regularity of $(\epsilon ,r)$-minimal surfaces, as presented in the previous version, follows from the foundational works [8] and [18]