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
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References

SHOWING 1-10 OF 28 REFERENCES
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
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3
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