A new functional for the Calculus of Variations, involving the variational mean curvature of sets in $R^n$

Abstract

We study the pth power $W_p(E)=\int _{{\mathbb R}^n}|H_E|^p\,dx$, where $H_E$ is the variational mean curvature of E, as defined in Barozzi (Rend Mat Acc Linceis 9(5):149–159, 1994) (see also in Proc Am Math Soc 99:313–316, 1987). This defines a functional $E\rightarrow W_p(E)$ on all bounded sets of finite perimeter. We prove that this functional is lower semicontinuous for every $p\ge 1$, with respect to the $L^1({\mathbb R}^n)$ topology. A corresponding compactness theorem also holds. The case $p=n$ appears to be particularly interesting. Finally, we introduce the pseudoconvex hull K of a bounded set $E\subset {\mathbb R}^n$ and establish the inequality $W_p(K)\le W_p(E)$.