For a smooth closed embedded planar curve Γ, we consider the minimization problem of the Willmore energy among immersed surfaces of a given genus 𝔤 ≥ 1 having the curve Γ as boundary, without any prescription on the conormal. In case Γ is a circle we prove that do not exist minimizers and that the infimum of the problem equals β𝔤 − 4π, where β𝔤 is the energy of the closed minimizing surface of genus 𝔤. We also prove that the same result also holds if Γ is a straight line for the suitable analogously defined minimization problem on asymptotically flat surfaces. Then we study the case in which Γ is compact, 𝔤 = 1 and the competitors are restricted to a suitable class 𝒞 of varifolds that includes embedded surfaces. We prove that under suitable assumptions minimizers exists in this class of generalized surfaces.
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DOI : 10.1051/cocv/2020049
Keywords: Willmore energy, Willmore surfaces with boundary, Navier boundary conditions, Simon’s ambient approach, existence
@article{COCV_2021__27_S1_A3_0,
author = {Pozzetta, Marco},
title = {On the {Plateau{\textendash}Douglas} problem for the {Willmore} energy of surfaces with planar boundary curves},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2020049},
mrnumber = {4222150},
zbl = {1467.49041},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2020049/}
}
TY - JOUR AU - Pozzetta, Marco TI - On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2020049/ DO - 10.1051/cocv/2020049 LA - en ID - COCV_2021__27_S1_A3_0 ER -
%0 Journal Article %A Pozzetta, Marco %T On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2020049/ %R 10.1051/cocv/2020049 %G en %F COCV_2021__27_S1_A3_0
Pozzetta, Marco. On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. S2. doi: 10.1051/cocv/2020049
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