Lower semicontinuity for the Helfrich problem

Sascha Eichmann

doi:10.1007/s10455-020-09718-5

Abstract

We minimise the Canham–Helfrich energy in the class of closed immersions with prescribed genus, surface area, and enclosed volume. Compactness is achieved in the class of oriented varifolds. The main result is a lower-semicontinuity estimate for the minimising sequence, which is in general false under varifold convergence by a counter example by Große-Brauckmann. The main argument involved is showing partial regularity of the limit. It entails comparing the Helfrich energy of the minimising sequence locally to that of a biharmonic graph. This idea is by Simon, but it cannot be directly applied, since the area and enclosed volume of the graph may differ. By an idea of Schygulla we adjust these quantities by using a two parameter diffeomorphism of {{mathbb {R}}}^3.

Highlights

This article deals with minimising the Helfrich energy for closed oriented smooth connected two-dimensional immersions f ∶ Σ → R3, which is defined as
H0 ∈ R is called spontaneous curvature. This energy was introduced by Helfrich [20] and Canham [3] to model the shape of blood cells
There lower semicontinuity was shown, but the limit has to be in C2, which is a priori not clear

Summary

Introduction

This article deals with minimising the Helfrich energy for closed oriented smooth connected two-dimensional immersions f ∶ Σ → R3 , which is defined as WH0 (f ) ∶= ∫Σ(H − H0) dμg. Vol(f) may become negativ dependend on the orientation Minimisers of such a problem satisfy the following Euler–Lagrange equation A ∈ R and V ∈ R correspond to Lagrange multipliers for the prescribed area and enclosed volume, respectively This differential equation is highly nonlinear since the Laplace-Beltrami Δf depends on the unknown immersion f. There lower semicontinuity was shown, but the limit has to be in C2 , which is a priori not clear This is a hard problem, since the Helfrich energy for oriented varifolds lacks a variational characterisation (cf (2.3) and cf [13, p. In Appendix A, we collect some usefull results for our reasoning

Compactness

First Variation of enclosed volume

Area and volume correction

Partial regularity and lower semicontinuity