The evolution of H-surfaces with a Plateau boundary condition
Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 109-157.

In this paper we consider the heat flow associated to the classical Plateau problem for surfaces of prescribed mean curvature. To be precise, for a given Jordan curve Γ 3 , a given prescribed mean curvature function H: 3 and an initial datum u o :B 3 satisfying the Plateau boundary condition, i.e. that u o | B :BΓ is a homeomorphism, we consider the geometric flow

t u-Δu=-2(Hu)D 1 u×D 2 uinB×(0,),
u(·,0)=u o onB,u(·,t)| B :BΓisweaklymonotoneforallt>0.
We show that an isoperimetric condition on H ensures the existence of a global weak solution. Moreover, we establish that these global solutions sub-converge as t to a conformal solution of the classical Plateau problem for surfaces of prescribed mean curvature.

@article{AIHPC_2015__32_1_109_0,
     author = {Duzaar, Frank and Scheven, Christoph},
     title = {The evolution of H-surfaces with a Plateau boundary condition},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {109--157},
     publisher = {Elsevier},
     volume = {32},
     number = {1},
     year = {2015},
     doi = {10.1016/j.anihpc.2013.10.003},
     zbl = {1328.53083},
     mrnumber = {3303944},
     language = {en},
     url = {www.numdam.org/item/AIHPC_2015__32_1_109_0/}
}
Duzaar, Frank; Scheven, Christoph. The evolution of H-surfaces with a Plateau boundary condition. Annales de l'I.H.P. Analyse non linéaire, Tome 32 (2015) no. 1, pp. 109-157. doi : 10.1016/j.anihpc.2013.10.003. http://www.numdam.org/item/AIHPC_2015__32_1_109_0/

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