The evolution of H-surfaces with a Plateau boundary condition

Abstract

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⊂R3, a given prescribed mean curvature function H:R3→R and an initial datum uo:B→R3 satisfying the Plateau boundary condition, i.e. that uo|∂B:∂B→Γ is a homeomorphism, we consider the geometric flow∂tu−Δu=−2(H∘u)D1u×D2uin B×(0,∞),u(⋅,0)=uoon B,u(⋅,t)|∂B:∂B→Γ is weakly monotone for all t>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.