Some aspects of the variational nature of mean curvature flow
The authors consider, in the first part of the paper, a second order space-time functional for which the solutions of the classical heat equation are minimizers. This point of view is the starting point for the study of a more difficult situation as the mean curvature flow of an embedded hypersurface in \(\mathbb R^n\). In particular the authors define an appropriate functional as the sum of two competing terms: the first depending on the normal \(\nu\) (singular in the part where \(\nu_{n+1}= 1\)) and the other, of second order, which is the Willmore functional of the time slice of the hypersurface integrated further in time. For the proposed functional the authors, with an appropriate definition of mean curvature flow, study, in a very extensive way, lower semicontinuity, coercivity, minimum problem and give, also, some results on approximation in the framework of the \(\Gamma\)-convergenge theory.
scientific article; zbMATH DE number 5365143