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

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.