Direct minimization of the Canham--Helfrich energy on generalized Gauss graphs

TL;DR

This paper proves the existence of minimizers for the Canham--Helfrich energy functional within the framework of generalized Gauss graphs, extending classical surface models and employing calculus of variations techniques.

Contribution

It extends the Canham--Helfrich functional to generalized Gauss graphs and establishes existence of minimizers under certain conditions, advancing the mathematical understanding of membrane energy models.

Findings

Existence of minimizers proved for generalized Gauss graphs.

Lower semicontinuity and compactness established under coerciveness conditions.

Remarks on regularity and behavior when coerciveness is absent.

Abstract

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.