Lower regularity assumption for an Euler-Lagrange equation on the contact line of the phase dependent Helfrich energy

TL;DR

This paper improves the regularity assumptions needed to derive the Euler-Lagrange equation on the phase separation line of the Helfrich energy, using a specialized test function involving the signed distance function.

Contribution

It lowers the regularity requirement for the phase separation line to C^{1,1} in deriving the Euler-Lagrange equation for the Helfrich energy.

Findings

Euler-Lagrange equation established on less regular phase separation line

Regularity assumption reduced to C^{1,1}

Method employs a test function with signed distance function

Abstract

We examine the phase dependent Helfrich energy and show an Euler-Lagrange equation on the phase seperation line. This result has already been observed by e.g. J\"ulicher-Lipowski and later Elliot-Stinner. Here we are able to lower the regularity assumption for this result down to $C^{1,1}$ for the seperation line. In the proof we employ a carefully choosen test function utilising the signed distance function.