# Stationary Surfaces with Boundaries

@article{Gruber2019StationarySW, title={Stationary Surfaces with Boundaries}, author={Anthony D. Gruber and Magdalena D Toda and Hung Tran}, journal={arXiv: Differential Geometry}, year={2019} }

The goal of this article is to investigate stationary surfaces with boundaries, which arise as critical points of functionals which depend on curvature. To that end, a generalized "bending energy" functional $\mathcal{W}$, involving a symmetric function in the principal curvatures, is considered. The first variation is computed, and a stress tensor is extracted, whose divergence quantifies deviation from $\mathcal{W}$-criticality. Boundary-value problems are then examined, and a… Expand

#### References

SHOWING 1-10 OF 37 REFERENCES

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

- Physics, Mathematics
- ESAIM: Control, Optimisation and Calculus of Variations
- 2021

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… Expand

Willmore obstacle problems under Dirichlet boundary conditions

- Mathematics
- 2021

We consider obstacle problems for the Willmore functional in the class of graphs of functions and surfaces of revolution with Dirichlet boundary conditions. We prove the existence of minimisers of… Expand

Connected surfaces with boundary minimizing the Willmore energy

- Mathematics
- Mathematics in Engineering
- 2020

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

On the variation of curvature functionals in a space form with application to a generalized Willmore energy

- Mathematics
- Annals of Global Analysis and Geometry
- 2019

Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and… 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

The Helfrich boundary value problem

- Mathematics
- 2018

We construct a branched Helfrich immersion satisfying Dirichlet boundary conditions. The number of branch points is finite. We proceed by a variational argument and hence examine the Helfrich energy… Expand

Uniform regularity results for critical and subcritical surface energies

- Physics, Mathematics
- Calculus of Variations and Partial Differential Equations
- 2018

We establish regularity results for critical points to energies of immersed surfaces depending on the first and the second fundamental form exclusively. These results hold for a large class of… Expand

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

- Mathematics
- Advances in Calculus of Variations
- 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

Nonuniqueness for Willmore Surfaces of Revolution Satisfying Dirichlet Boundary Data

- Mathematics
- 2016

In this note Willmore surfaces of revolution with Dirichlet boundary conditions are considered. We show two nonuniqueness results by reformulating the problem in the hyperbolic half plane and solving… Expand