Non-parametric hypersurfaces of prescribed mean curvature in H n x R

In the present thesis, we consider hypersurfaces of prescribed mean curvature and prescribed boundary values in the product manifold $HypimesR$ from a variational point of view. Here, $Hyp$ denotes the conformal ball model of the hyperbolic n-space. We focus mainly on non-parametric hypersurfaces but also establish a link to the parametric formulation of the corresponding Dirichlet problem in order to use techniques from geometric measure theory. The non-parametric hypersurfaces are given by graphs of height functions u over a given bounded (with respect to the hyperbolic metric) domain in $Hyp$.We solve a certain Dirichlet problem for the function u in the framework of the direct methods of the calculus of variation. The corresponding functional to be minimized is given byegin{align*}egin{aligned}J(u) &:=&&intlimits_OmegaHypvolsqrt{1+Hypvolis|Du|^2} + intlimits_Omega Hypvol intlimits_0^{u(x)}H(x,t),dtdx\& && + ointlimits_{partialOmega} Hypvolm|u-varphi|,dHL^{n-1},.end{aligned}end{align*}The functional J is well defined on functions $uinBVH(Omega)$, where $OmegaceHyp$ with partialOmegain C^0,1 is a bounded domain. The boundary values $varphiinLH(partialOmega)$ and the mean curvature Hin C^0,1(overline{Omega}imesR) with $frac{partial}{partial t} H(x,t) > 0$ are given.We establish existence and uniqueness of a minimizer u to J. Furthermore, we prove height bounds for u if $varphi in LI(Omega)$. The interior regularity of the minimizer is proved via regularity theory of rectifiable $lambda$-minimizing currents from geometric measure theory up to dimension $n leq 6$. For the proof of the interior regularity of the minimizer in arbitrary dimension $ngeq 2$ we use regularity theory of quasilinear, non-strict uniformly elliptic partial differential equations. As an addition, we give an introduction to the space of functions of bounded variation on Riemannian manifolds.

Grantor dc:publisher

Publikationsserver der RWTH Aachen University

Year dc:date

2010

Author dc:creator

• Roeser, Frank

Contributors dc:contributor

• von der Mosel, Heiko

• info:eu-repo/classification/ddc/510
• Analysis
• Geometrische Maßtheorie
• Variationsrechnung
• Partielle Differentialgleichung
• Mathematik
• geometric measure theory
• calculus of variations
• partial differential equations

Statement dc:rights

• info:eu-repo/semantics/openAccess

Language dc:language

eng

Identifier

https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114077%22

OAI record oai:identifier

https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A51826