Free boundary problems governed by mean curvature

by Alexander Volkmann

Institution: Freie Universität Berlin
Department: FB Mathematik und Informatik
Degree: PhD
Year: 2015
Record ID: 1109906
Full text PDF: http://edocs.fu-berlin.de/diss/receive/FUDISS_thesis_000000098728


In this thesis we consider the following three free boundary value problems for (hyper-)surfaces that are governed by the mean curvature of the (hyper-)surface: 1. "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 the unit ball in Rn that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting, thereby generalizing results of Oliveira and Soret, and Fraser and Schoen. Then we derive some sharp geometric inequalities for compact surfaces with free boundaries inside arbitrary orientable support surfaces of class C2. Furthermore, we obtain a sharp lower bound for the L1-tangent-point energy of closed curves in R3 thereby answering a question raised by Strzelecki, Szumańska, and von der Mosel. 2. "Relative isoperimetric properties of asymptotically flat support surfaces" We define a notion of mass for asymptotically flat hypersurfaces S of euclidean space and prove a positive mass theorem in all dimensions. Then we establish a free boundary version of an obstruction discovered by Schoen and Yau in their proof of the positive mass theorem, and refined by Eichmair and Metzger, and very recently by Carlotto: positive mean curvature of S in R3 is not compatible with the existence of (certain) stable free boundary minimal surfaces. We then use this to prove that given a compact set K of R3, all volume-preserving stable free boundary constant mean curvature surfaces with respect to S of sufficiently large boundary length will avoid K, thereby obtaining a free boundary version of the main result in [Eichmair-Metzger, 2012]. Finally, inspired by ideas of Eichmair and Metzger we prove the existence of arbitrarily large isoperimetric regions relative to S. 3. "Weak solutions of nonlinear mean curvature flow with Neumann boundary condition" We propose a new flow approach to obtain relative isoperimetric inequalities. As a first step in this program we develop a weak level set formulation for mean curvature flow and positive powers of mean curvature flow with Neumann boundary condition. We prove the existence of weak solution under natural conditions on the supporting surface and derive some properties for the evolving surfaces. The case of surfaces without boundary has been treated by Schulze. In der vorliegenden Arbeit betrachten wir drei freie Randwertprobleme für (Hyper-)Flächen welche durch die mittlere Krümmung der (Hyper-)Fläche beschrieben werden: 1. "Eine Monotonieformel für Flächen mit freiem Rand bezüglich der Einheitskugel" Wir beweisen eine Monotonieidentität für kompakte Flächen mit freien Rändern in dem Rand der Einheitskugel des Rn welche quadratisch integrierbare mittlere Krümmung besitzen. Als eine Konsequenz erhalten wir eine Ungleichung vom Li-Yau Typ für diesen Fall, wodurch wir Resultate von Oliveira und Soret, und Fraser und Schoen verallgemeinern. Im Anschluss leiten wir einige scharfe geometrische…