Volume 27, 2021Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
|Number of page(s)||35|
|Published online||01 March 2021|
On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves
Dipartimento di Matematica, Università di Pisa,
Largo Bruno Pontecorvo 5,
* Corresponding author: firstname.lastname@example.org
Accepted: 15 July 2020
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.
Mathematics Subject Classification: 49J40 / 49J45 / 49Q20 / 53A05 / 49Q15 / 53A30
Key words: Willmore energy / Willmore surfaces with boundary / Navier boundary conditions / Simon’s ambient approach / existence
© EDP Sciences, SMAI 2021
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