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Daily Overview |
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Gvp2: Geometric variational problems Location: A704 Session Chair: Fabian Rupp Session Chair: Christian Scharrer | |
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Driving a Wedge in a Willmore Half-Sphere 1: Universität Augsburg, Germany; 2: Universität Wien, Austria We minimize the Willmore energy among topological half-spheres that satisfy
Standard questions for such obstacle problems include existence and optimal regularity of minimizers as well as the size of the coincidence set, i.e., the set where the obstacle is touched. Existence theory is obstructed by Möbius invariance of the Willmore energy. However, some modifications of known compactness results resolve this issue. After discussing existence, we focus on conic obstacles (with some mild restrictions on their opening angle). Our main observation is that these obstacles are touched (i) nowhere if the cone lies below the round half-sphere and (ii) only once at the tip of the cone if not. In case (i) the minimizer is given by the round half sphere. In case (ii) we obtain that minimizers must satisfy the Willmore equation with a Dirac measure on the right hand side. This implies that minimizers lie in W^{3,p} for any p < 2 but not in W^{3,2}, as we will discuss. | |
