On the Plateau-Douglas problem for the Willmore energy of surfaces with planar boundary curves

Abstract

For a smooth closed embedded planar curve $\Gamma$, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus $\mathfrak{g}\ge 1$ having the curve $\Gamma$ as boundary, without any prescription on the conormal. By general lower bound estimates, in case $\Gamma$ is a circle we prove that such problem is equivalent if restricted to embedded surfaces, we prove that do not exist minimizers, and the infimum equals ${\beta }_{\mathfrak{g}}-4\pi$, where ${\beta }_{\mathfrak{g}}$ is the energy of the closed minimizing surface of genus $\mathfrak{g}$. We also prove that the same result also holds if $\Gamma$ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces.\\ Then we study the case in which $\Gamma$ is compact, $\mathfrak{g}=1$ and the competitors are restricted to a suitable class $\mathcal{C}$ of varifolds including embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces.

References in corpus (4)

• Existence and Regularity of Spheres Minimising the Canham-Helfrich Energy
• A Resolution of the Poisson Problem for Elastic Plates
• The Helfrich Boundary Value Problem
• Connected surfaces with boundary minimizing the Willmore energy

Cited by in corpus (2)

• Connected surfaces with boundary minimizing the Willmore energy
• Degenerate elastic networks