Local solutions to a free boundary problem for the Willmore functional

  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},
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
On the Plateau–Douglas problem for the Willmore energy of surfaces with planar boundary curves
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
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
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 boundaryExpand
Existence for Willmore surfaces of revolution satisfying non-symmetric Dirichlet boundary conditions
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 admissibleExpand
Reflection of Willmore Surfaces with Free Boundaries
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 planeExpand
Uniform regularity and convergence of phase-fields for Willmore’s energy
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 uniformExpand
Radial symmetry for p-harmonic functions in exterior and punctured domains
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 itsExpand
On the boundary regularity of phase-fields for Willmore's energy
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 beExpand
Large area-constrained Willmore surfaces in asymptotically Schwarzschild 3-manifolds
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 suchExpand
Rigidity and stability of spheres in the Helfrich model
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 thatExpand


A monotonicity formula for free boundary surfaces with respect to the unit ball
We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in $\mathbb R^n$ that have square integrable mean curvature. As one consequence we obtain aExpand
Concentration of small Willmore spheres in Riemannian 3-manifolds
Given a 3-dimensional Riemannian manifold $(M,g)$, we prove that if $(\Phi_k)$ is a sequence of Willmore spheres (or more generally area-constrained Willmore spheres), having Willmore energy boundedExpand
Boundary value problems for variational integrals involving surface curvatures
The following investigation deals with surfaces governed by and extremal for a free energy functional which is quadratic in the principal curvatures. The associated Euler-Lagrange differentialExpand
Boundary value problems for the one-dimensional Willmore equation
The one-dimensional Willmore equation is studied under Navier as well as under Dirichlet boundary conditions. We are interested in smooth graph solutions, since for suitable boundary data, we expectExpand
Some results about the existence of critical points for the Willmore functional
Using a perturbative approach, it is shown existence and multiplicity of critical points for the Willmore functional in ambient manifold $${(\mathbb{R}^3, g_\epsilon)}$$ —where $${g_\epsilon}$$ is aExpand
Elliptic Partial Differential Equations of Second Order
We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equationsExpand
Uniqueness for the homogeneous Dirichlet Willmore boundary value problem
We give a sufficient condition for curves on a plane or on a sphere such that if these give the boundary of a Willmore surface touching tangentially along the boundary the plane or the sphereExpand
Classical solutions to the Dirichlet problem for Willmore surfaces of revolution
Abstract We consider the Willmore equation with Dirichlet boundary conditions for a surface of revolution obtained by rotating the graph of a positive smooth even function. We show existence of aExpand
A duality theorem for Willmore surfaces
so the two functional differ by a constant. The functional i^(X) has the advantage that its integrand is nonnegative and vanishes exactly at the umbilic points of the immersion X. Obviously iT(X) = 0Expand
Complete Willmore surfaces in H3 with bounded energy: boundary regularity and bubbling
We study various aspects related to boundary regularity of complete properly embedded Willmore surfaces in H3, particularly those related to assumptions on boundedness or smallness of a certainExpand