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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Weak solutions to the heat flow for surfaces of prescribed mean curvature


Authors: Verena Bögelein, Frank Duzaar and Christoph Scheven
Journal: Trans. Amer. Math. Soc. 365 (2013), 4633-4677
MSC (2010): Primary 53A10, 58J35; Secondary 35K51, 35B40
Published electronically: April 9, 2013
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Abstract: In this paper we establish the existence of global weak solutions to the heat flow for surfaces of prescribed mean curvature, i.e. the existence for the Cauchy-Dirichlet problem to parabolic systems of the type

$\displaystyle \left \{ \begin {array}{c} \partial _t u-\Delta u =-2 (H\circ u)D... ...on $\partial _{\rm par} \big (B\times (0,\infty )\big )$}, \end{array} \right .$    

where $ H\colon \mathbb{R}^3\to R$ is a bounded continuous function satisfying an isoperimetric condition, $ B$ is the unit ball in $ \mathbb{R}^2$ and $ u\colon B\times (0,\infty )\to \mathbb{R}^3$. As one of the possible applications we show that the problem has a solution with values in $ B_R\subset \mathbb{R}^3$, whenever $ u_o(B)\subseteq B_R$ and furthermore there holds

$\displaystyle \int _{\{ \xi \in B_R: \vert H(\xi )\vert\ge \frac {3}{2R}\}}\vert H\vert^3\, d\xi <\frac {9\pi }{2}, \qquad \vert H(a)\vert\le \tfrac {1}{R}$$\displaystyle \quad \mbox {for $a\in \partial B_R$.}$    


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Additional Information

Verena Bögelein
Affiliation: Department Mathematik, Universität Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
Email: boegelein@math.fau.de

Frank Duzaar
Affiliation: Department Mathematik, Universität Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
Email: duzaar@math.fau.de

Christoph Scheven
Affiliation: Department Mathematik, Universität Erlangen–Nürnberg, Cauerstrasse 11, 91058 Erlangen, Germany
Address at time of publication: Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, 47057 Duisburg, Germany
Email: scheven@mi.uni-erlangen.de, christoph.scheven@uni-due.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05885-0
PII: S 0002-9947(2013)05885-0
Keywords: Surfaces of prescribed mean curvature, gradient flow, global solutions, asymptotic behavior
Received by editor(s): June 3, 2011
Published electronically: April 9, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.