No abstract

We prove that, in the flat torus and in any dimension, the volume-preserving mean curvature flow and the surface diffusion flow, starting C 1,1 −close to a strictly stable critical set of the perimeter E, exist for all times and converge to a translate of E exponentially fast as time goes to infinity.

The existence of minimizers of the Canham-Helfrich functional in the setting of generalized Gauss graphs is proved. As a first step, the Canham-Helfrich functional, usually defined on regular surfaces, is extended to generalized Gauss graphs, then lower semicontinuity and compactness are proved under a suitable condition on the bending constants ensuring coerciveness; the minimization follows by the direct methods of the Calculus of Variations. Remarks on the regularity of minimizers and on the behavior of the functional in case there is lack of coerciveness are presented.