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December 4, 2009
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We prove some estimates and regularity results for oblique derivative problems on Reifenberg flat domains with not very smooth coefficients and vectorfields by transforming these problems into conormal ones.
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December 4, 2009
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If G ⊂ ℝ 2 is either a bounded or an exterior domain weak L q -solutions of Δ w = rot ∗ p in G , w | ∂ G = 0 ( p ∈ L q ( G ) given), are regarded. Because of the simple relation between rot and div in case n = 2 we deduce immediately from the corresponding results for Δ v = ∇ p in G , v | ∂ G = 0, all results for the problem considered above. In addition we derive a complete characterization of all solenoidal vector fields. This depends on topological properties of G .
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December 4, 2009
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In this article the author firstly gives a short overview over the results which have been achieved so far concerning the question which boundary contours in ℝ 3 can bound only finitely many (stable) immersed minimal surfaces. After that the author describes the course of the proof of his result in [22], which states that a simple closed polygon in ℝ 3 can bound only finitely many immersed minimal surfaces of disc-type if it meets the following two requirements: firstly it has to bound only minimal surfaces without boundary branch points, and secondly its total curvature, i.e. the sum of the exterior angles at its vertices, has to be smaller than 6 π .
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September 25, 2009
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We extend to objects of general convex form an earlier criterion given by Finn for ensuring a local energy minimum corresponding to floating of a heavy horizontal cylinder on a liquid surface. We obtain a formal analogue of Finn´s criterion, and additionally a condition on interfacial surface energies under which an object of given shape with density exceeding that of the liquid must float if it is scaled to be sufficiently small. In analogy with Finn´s earlier work, we show also in the more physical case presently considered that if a body is lowered rigidly from a position above the undisturbed liquid surface to a position below the surface, the liquid configuration must change during the motion in a discontinuous way.
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December 4, 2009
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In this paper, a new result dealing with high indices theorems has been obtained.
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December 4, 2009
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In this article it will be proven that there are surfaces that become singular under the Willmore flow. For this purpose, we investigate the Willmore flow of radially symmetric immersions of the sphere using a blow-up construction due to E. Kuwert and R. Schätzle. It will be shown that in this situation the blow-up limit is a surface of revolution as well and is either a round sphere or consists of planes and catenoids. Furthermore, we give an estimate for the number of these planes and catenoids in terms of the Willmore energy of the initial surface. This will enable us to show that there are immersions of the sphere with a Willmore energy arbitrarily close to 8 π that do not converge to a Willmore immersion under the Willmore flow. Either a small quantum of the curvature concentrates or the diameter of the surface does not stay bounded under the Willmore flow.
