Weak solutions to the heat flow for surfaces of prescribed mean curvature
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- by Verena Bögelein, Frank Duzaar and Christoph Scheven PDF
- Trans. Amer. Math. Soc. 365 (2013), 4633-4677 Request permission
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 \begin{equation*} \left \{ \begin {array}{c} \partial _t u-\Delta u =-2 (H\circ u)D_1u\times D_2u\quad \mbox {in $B\times (0,\infty )$,}\\[3pt] u=u_o\quad \mbox {on $\partial _\textrm {par} \big (B\times (0,\infty )\big )$}, \end{array} \right . \end{equation*} 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 \begin{equation*} \int _{\{ \xi \in B_R: |H(\xi )|\ge \frac {3}{2R}\}}|H|^3 d\xi <\frac {9\pi }{2}, \qquad |H(a)|\le \tfrac {1}{R}\quad \mbox {for $a\in \partial B_R$.} \end{equation*}References
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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
- Received by editor(s): June 3, 2011
- Published electronically: April 9, 2013
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 365 (2013), 4633-4677
- MSC (2010): Primary 53A10, 58J35; Secondary 35K51, 35B40
- DOI: https://doi.org/10.1090/S0002-9947-2013-05885-0
- MathSciNet review: 3066767