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

TL;DR

This work extends the Canham--Helfrich energy from smooth membranes to generalized Gauss graphs, proving existence of minimizers under a coercivity condition $4\alpha_H>\alpha_K>0$ by establishing lower semicontinuity and compactness and applying the direct method. The energy is formulated via a convex standard integrand $\tilde{f}$ on the Gauss-graph data, ensuring consistency with the classical energy on smooth surfaces. The authors obtain minimizers in several unconstrained and area/volume-constrained classes, and show that minimizers induce $\mathcal{C}^2$-rectifiable 2D supports, thereby connecting generalized-graph minimizers to traditional curvature-varifold regularity. Overall, the results provide a rigorous variational framework that accommodates physically relevant parameter ranges while guaranteeing well-posed existence and meaningful regularity for membrane models.

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.

Theorems & Definitions (37)

• Theorem 2.1: Federer
• Definition 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Lemma 3.1: LR
• proof
• Remark 3.2
• Lemma 3.3: LR
• proof