{"id":{"repo_id":"aachen","oai_identifier":"oai:publications.rwth-aachen.de:51826"},"canonical_url":"https://search.dev.ndltd.org/etd/aachen/oai:publications.rwth-aachen.de:51826","repository":{"repo_id":"aachen","name":"RWTH Aachen University","base_url":"https://publications.rwth-aachen.de/oai2d"},"display":{"title":"Non-parametric hypersurfaces of prescribed mean curvature in H n x R","abstract":"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.","abstract_html":"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) &amp;:=&amp;&amp;intlimits_OmegaHypvolsqrt{1+Hypvolis|Du|^2} + intlimits_Omega Hypvol intlimits_0^{u(x)}H(x,t),dtdx\\&amp; &amp;&amp; + 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 <span class=\"etd-inline-math\">partialOmegain C<sup>0,1</sup></span> is a bounded domain. The boundary values $varphiinLH(partialOmega)$ and the mean curvature <span class=\"etd-inline-math\">Hin C<sup>0,1</sup>(overline{Omega}imesR)</span> with $frac{partial}{partial t} H(x,t) &gt; 0$ 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.","abstract_has_math":true,"creators":["Roeser, Frank"],"institution":"Publikationsserver der RWTH Aachen University","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["von der Mosel, Heiko"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010","date_published":"2010","updated_at":"2026-07-30T19:40:42Z","subjects":["info:eu-repo/classification/ddc/510","Analysis","Geometrische Maßtheorie","Variationsrechnung","Partielle Differentialgleichung","Mathematik","geometric measure theory","calculus of variations","partial differential equations"],"languages":["eng"],"rights":["info:eu-repo/semantics/openAccess"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114077%22"],"render_values":[{"text":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114077%22","href":"https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114077%22","code":true}]}]},"links":{"outbound_url":"https://publications.rwth-aachen.de/record/51826","outbound_label":"Repository record","outbound_source":"dc:identifier"},"source_record":{"url":"https://publications.rwth-aachen.de/oai2d?verb=GetRecord&metadataPrefix=oai_dc&identifier=oai%3Apublications.rwth-aachen.de%3A51826","prefix":"oai_dc"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["von der Mosel, Heiko"]},{"key":"dc:creator","label":"Author","values":["Roeser, Frank"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:coverage","label":"Dc Coverage","values":["DE"]},{"key":"dc:date","label":"Dc Date","values":["2010"]},{"key":"dc:publisher","label":"Institution","values":["Publikationsserver der RWTH Aachen University"]},{"key":"dc:relation","label":"Dc Relation","values":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-32941"]},{"key":"dc:type","label":"Dc Type","values":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["info:eu-repo/classification/ddc/510","Analysis","Geometrische Maßtheorie","Variationsrechnung","Partielle Differentialgleichung","Mathematik","geometric measure theory","calculus of variations","partial differential equations"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["info:eu-repo/semantics/openAccess"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["https://publications.rwth-aachen.de/record/51826","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114077%22"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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."]},{"key":"dc:source","label":"Dc Source","values":["Aachen : Publikationsserver der RWTH Aachen University X, 148 S. : graph. Darst. (2010). = Aachen, Techn. Hochsch., Diss., 2010"]},{"key":"dc:title","label":"Title","values":["Non-parametric hypersurfaces of prescribed mean curvature in H n x R"]}]}],"canonical_facts":{"dc:contributor":["von der Mosel, Heiko"],"dc:coverage":["DE"],"dc:creator":["Roeser, Frank"],"dc:date":["2010"],"dc:description":["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."],"dc:identifier":["https://publications.rwth-aachen.de/record/51826","https://publications.rwth-aachen.de/search?p=id:%22RWTH-CONV-114077%22"],"dc:language":["eng"],"dc:publisher":["Publikationsserver der RWTH Aachen University"],"dc:relation":["info:eu-repo/semantics/altIdentifier/urn/urn:nbn:de:hbz:82-opus-32941"],"dc:rights":["info:eu-repo/semantics/openAccess"],"dc:source":["Aachen : Publikationsserver der RWTH Aachen University X, 148 S. : graph. Darst. (2010). = Aachen, Techn. Hochsch., Diss., 2010"],"dc:subject":["info:eu-repo/classification/ddc/510","Analysis","Geometrische Maßtheorie","Variationsrechnung","Partielle Differentialgleichung","Mathematik","geometric measure theory","calculus of variations","partial differential equations"],"dc:title":["Non-parametric hypersurfaces of prescribed mean curvature in H n x R"],"dc:type":["info:eu-repo/semantics/doctoralThesis","info:eu-repo/semantics/publishedVersion"]},"updated_at":"2026-07-30T19:40:42Z"}