Inhaltsverzeichnis - Mathematisches Institut der Universität zu Köln

Anna Dall’Acqua
Otto-von-Guericke-Universität Magdeburg
Willmore surfaces with boundary
DMV Tagung 2011 - Köln, 19. - 22. September
The Willmore functional associates to a surface the integral, over the surface, of its mean curvature squared.
Critical points of this functional are called Willmore surfaces and are solutions of a fourth order
non-linear elliptic p.d.e., the Willmore equation. Examples of Willmore surfaces are spheres and minimal
surfaces. The Willmore equation may be considered as a frame invariant counterpart of the clamped plate
equation. This equation is of interest not only in mechanics and membrane physics but also in differential
geometry.
The problem we are interested in is the existence of Willmore surfaces which obey suitable boundary conditions.
Being the Willmore equation of fourth order, one needs to impose two sets of boundary conditions
on the boundary. In this talk we give an overview on results concerning existence, regularity, uniqueness
and stability properties of Willmore surfaces satisfying Dirichlet or natural boundary conditions in certain
symmetric situations. We use methods from the Calculus of Variations.
Literatur
Bergner, M., Dall’Acqua, A., Fröhlich, S. (2010). Willmore surfaces of revolution with two prescribed boundary
circles, to appear in The Journal of Geometric Analysis.
Dall’Acqua, A. (2011) Uniqueness for the homogeneous Dirichlet Willmore boundary value problem, preprint.
Dall’Acqua, A., Fröhlich, S., Grunau, H.-Ch., Schieweck, F. (2011). Symmetric Willmore Surfaces of revolution
satisfying arbitrary Dirichlet boundary data Advances in Calculus of Variations, 4, 1–81.
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