The Helfrich Boundary Value Problem

TL;DR

This paper establishes a lower-semicontinuity result for the Helfrich energy in the context of oriented varifolds, enabling the construction of branched Helfrich immersions satisfying boundary conditions.

Contribution

It introduces a variational approach to the Helfrich boundary value problem and proves a key lower-semicontinuity result for the energy functional.

Findings

Constructed a branched Helfrich immersion with finite branch points.

Proved lower-semicontinuity of Helfrich energy for varifold convergence.

Identified limitations of energy lower-semicontinuity with arbitrary sequences.

Abstract

We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy for oriented varifolds. The main contribution of this paper is a lower-semicontinuity result with respect to oriented varifold convergence for the Helfrich energy and a minimising sequence. For arbitrary sequences this is false by a counterexample of Gro{\ss}e-Brauckmann.