Calculus of Variations and Geometric Measure Theory
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M. Pozzetta

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

created by pozzetta1 on 17 Oct 2018
modified on 16 Jul 2021


Accepted Paper

Inserted: 17 oct 2018
Last Updated: 16 jul 2021

Journal: ESAIM: COCV
Year: 2021


For a smooth closed embedded planar curve $\Gamma$, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus $\mathfrak{g}\geq1$ 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.


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