Calculus of Variations and Geometric Measure Theory

J. Dalphin - A. Henrot - S. Masnou - T. Takahashi

On the minimization of total mean curvature

created by masnou on 19 Feb 2015


Submitted Paper

Inserted: 19 feb 2015
Last Updated: 19 feb 2015

Year: 2014


In this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partial\Omega} H\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ from convex smooth sets to any regular open set $\Omega\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature of $\partial\Omega$ and $A$ the area. We prove that this inequality holds true for axisymmetric domains which are convex in the direction orthogonal to the axis of symmetry. We also show that this inequality cannot be true in more general situations. However we prove that $\int_{\partial\Omega}
\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ remains true for any axisymmetric domain.