Sobolev trace inequalities on John domains and its applications

Weicong Su; Yi Ru-Ya Zhang

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arXiv · 2406.06906

Sobolev trace inequalities on John domains and its applications

Weicong Su·Yi Ru-Ya Zhang

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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}.

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Weicong Su, Yi Ru-Ya Zhang. 2024-06-11. Sobolev trace inequalities on John domains and its applications. https://arxiv.org/abs/2406.06906

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