Sobolev trace inequalities on John domains and its applications

We prove that a trace inequality holds for John domains $\mathrm{\Omega }$ satisfying $${\mathcal{H}}^{n-1}(\mathrm{\partial }\mathrm{\Omega }\setminus {\mathrm{\partial }}_{\ast }\mathrm{\Omega })=0,$$ where ${\mathrm{\partial }}_{\ast }\mathrm{\Omega }$ denotes the measure-theoretic boundary, together with an upper density bound on $\mathrm{\partial }\mathrm{\Omega }$. 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 $\mathrm{\partial }\mathrm{\Omega }$ 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}.