# Local solutions to a free boundary problem for the Willmore functional

@article{Alessandroni2014LocalST, title={Local solutions to a free boundary problem for the Willmore functional}, author={R. Alessandroni and E. Kuwert}, journal={Calculus of Variations and Partial Differential Equations}, year={2014}, volume={55}, pages={1-29} }

We consider a free boundary problem for the Willmore functional $$\mathcal{W}(f) = \frac{1}{4} \int _\Sigma H^2\,d\mu _f$$W(f)=14∫ΣH2dμf. Given a smooth bounded domain $$\Omega \subset {\mathbb R}^3$$Ω⊂R3, we construct Willmore disks which are critical in the class of surfaces meeting $$\partial \Omega $$∂Ω at a right angle along their boundary and having small prescribed area. Using rescaling and the implicit function theorem, we first obtain constrained solutions with prescribed barycenter on… Expand

#### 13 Citations

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

- Mathematics
- 2018

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 $\Gamma$… Expand

A Resolution of the Poisson Problem for Elastic Plates

- Physics, Mathematics
- 2018

We consider the problem of finding a surface $$\Sigma \subset {\mathbb {R}}^m$$ Σ ⊂ R m of least Willmore energy among all immersed surfaces having the same boundary, boundary Gauss map and area.… Expand

Connected surfaces with boundary minimizing the Willmore energy

- Mathematics
- 2019

For a given family of smooth closed curves $\gamma^1,...,\gamma^\alpha\subset\mathbb{R}^3$ we consider the problem of finding an elastic \emph{connected} compact surface $M$ with boundary… Expand

Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions

- Mathematics
- 2017

Abstract In this paper, existence for Willmore surfaces of revolution is shown, which satisfy non-symmetric Dirichlet boundary conditions, if the infimum of the Willmore energy in the admissible… Expand

Reflection of Willmore Surfaces with Free Boundaries

- Mathematics
- Canadian Journal of Mathematics
- 2020

Abstract We study immersed surfaces in
${\mathbb R}^3$
that are critical points of the Willmore functional under boundary constraints. The two cases considered are when the surface meets a plane… Expand

Uniform regularity and convergence of phase-fields for Willmore’s energy

- Mathematics
- 2015

We investigate the convergence of phase fields for the Willmore problem away from the support of a limiting measure $$\mu $$μ. For this purpose, we introduce a suitable notion of essentially uniform… Expand

Radial symmetry for p-harmonic functions in exterior and punctured domains

- Mathematics, Physics
- 2018

ABSTRACT We prove symmetry for the p-capacitary potential satisfying under Serrin’s overdetermined condition Here is any bounded domain on which no a priori assumption is made, and denotes its… Expand

On the boundary regularity of phase-fields for Willmore's energy

- Mathematics
- Proceedings of the Royal Society of Edinburgh: Section A Mathematics
- 2019

Abstract We demonstrate that Radon measures which arise as the limit of the Modica-Mortola measures associated with phase-fields with uniformly bounded diffuse area and Willmore energy may be… Expand

Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds

- Mathematics
- 2021

We apply the method of Lyapunov-Schmidt reduction to study large area-constrained Willmore surfaces in Riemannian 3-manifolds asymptotic to Schwarzschild. In particular, we prove that the end of such… Expand

Rigidity and stability of spheres in the Helfrich model

- Physics, Mathematics
- 2014

The Helfrich functional, denoted by H^{c_0}, is a mathematical expression proposed by Helfrich (1973) for the natural free energy carried by an elastic phospholipid bilayer. Helfrich theorises that… Expand

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