Minkowski Inequalities via Nonlinear Potential Theory

By Virginia Agostiniani, Mattia Fogagnolo, L. Mazzieri. Archive for Rational Mechanics and Analysis, 2019

51 citations recorded by the source corpus.

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set Ω⊂Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^n$$\end{document}, n≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 3$$\end{document}. Our proof relies on the discovery of effective monotonicity formulas holding along the level set flow of the p-capacitary potentials associated with Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, for every p sufficiently close to 1. These formulas also testify the existence of a link between the monotonicity formulas derived by Colding and Minicozzi for the level set flow of Green’s functions and the monotonicity formulas employed by Huisken, Ilmanen and several other authors in studying the geometric implications of the Inverse Mean Curvature Flow. In dimension n≥8\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 8$$\end{document}, our conclusions are stronger than the ones obtained so far through the latter mentioned technique.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/9869352810c4da88ebf150a9892ff342defefc0c

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In Applied Mathematics

  1. Mattia Fogagnolo
  2. L. Mazzieri
  3. Virginia Agostiniani
  4. Zoltán M. Balogh
  5. Alexandru Krist'aly
  6. M. Rigoli

In Mathematical Physics

  1. L. Mazzieri
  2. Mattia Fogagnolo
  3. Virginia Agostiniani

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