Minimisers of Helfrich functional for surfaces of revolution

Abstract

<p style='text-indent:20px;'>In this paper we investigate the existence and the properties for the minimisers of a special Helfrich functional for surfaces of revolution with Dirichlet boundary value conditions. Removing the even restriction for the admissible functions in [<xref ref-type="bibr" rid="b5">5</xref>], we prove that the minimiser is even and smooth, the minimal increases as the boundary value increases, and the minimiser is no less than the boundary value which answers an open question in [<xref ref-type="bibr" rid="b5">5</xref>] partly. We also obtain the existence and regularity for (general) Helfrich functional when the boundary value is large.</p>