The mean curvature of a Lipschitz continuous manifold

Barozzi, Elisabetta   ; Gonzalez, Eduardo   ; Massari, Umberto

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 4, pp. 257-277

Résumé

The paper is devoted to the description of some connections between the mean curvature in a distributional sense and the mean curvature in a variational sense for several classes of non-smooth sets. We prove the existence of the mean curvature measure of $\partial E$ by using a technique introduced in [4] and based on the concept of variational mean curvature. More precisely we prove that, under suitable assumptions, the mean curvature measure of $\partial E$ is the weak limit (in the sense of distributions) of the mean curvatures of a sequence of regular $n$-dimensional manifolds $M_{j}$ convergent to $\partial E$. The manifolds $M_{j}$ are closely related to the level surfaces of the variational mean curvature $H_{E}$ of $E$.