Boundary Value Problems for Evolutions of Willmore Type

Geometric gradient flows of energy functionals involving the curvature of a given object have become an indispensable tool both to understand problems in pure mathematics and to model a wide range of phenomena in the natural sciences. A prominent example of such a functional is the Willmore energy which is given by the integrated squared mean curvature of the given surface. Its L²-gradient flow, the Willmore flow, is a parabolic quasilinear evolution law of fourth order. While there is an abundance of results on the behaviour of curves and closed surfaces, there remains a huge number of open questions regarding boundary value problems for the Willmore flow. The first part of the thesis is devoted to the Willmore flow of two- or higher-dimensional immersed compact open hypersurfaces in Euclidean space with Navier boundary conditions which demand the boundary to remain fixed during the evolution and the mean curvature to vanish on the boundary. We initiate the research on this flow showing existence of strong solutions in anisotropic Sobolev spaces given a sufficiently smooth initial surface that has zero mean curvature on the boundary and is close to an appropriate reference manifold. The regularity of the initial immersion corresponds to the trace space of the solution space. Considering motions that are given as graphs over the fixed reference geometry we may write the evolution in terms of a scalar function describing the position of the evolving surface with respect to the reference manifold. The required analysis is technically elaborate as the evolution needs to be translated to local charts. In the second part of the thesis we study planar networks composed of three immersed curves that meet in one or two triple junctions and may or may not have endpoints fixed in the plane. The elastic energy of such a configuration is given by the sum of the Willmore energies of the single curves each containing a positively weighted length penalisation term. Its L²-gradient flow leads to a system of Willmore type evolution laws with natural nonlinear coupled boundary conditions. Hereby, the curves need to stay attached but the junctions are allowed to move. The major difficulties lie in the tangential degrees of freedom which are due to geometric nature of the problem. We show existence of solutions in the strong and classical sense, namely in anisotropic Sobolev spaces and parabolic Holder spaces. In both cases compatibility and regularity assumptions on the initial network are required. We further establish uniqueness of solutions in both function space settings in a purely geometric sense showing that any two observable motions solving the flow are reparametrisations of each other. The parabolic nature of the problem allows us to show in addition that solutions are smooth for positive times. As a main result we show that the flow exists globally in time if the length of each curve remains uniformly bounded away from zero and if at least one angle at the triple junction stays uniformly bounded away from zero, π and 2π . The proof relies on energy estimates and the existence of solutions in the Sobolev setting on time intervals of uniform length quantifiable in terms of the initial network.