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Convergence of surfaces of prescribed mean curvature. (English) Zbl 0696.49070
Let \(\Omega\) be the unit disk in \({\mathbb{R}}^ 2\), \(\Omega =\{(\alpha,\beta)\in {\mathbb{R}}^ 2|\) \(\alpha^ 2+\beta^ 2<1\}\), let \(\Gamma\) be a Jordan curve included in a ball of radius R of \({\mathbb{R}}^ 3\), and \(H\in C^{\infty}({\mathbb{R}}^ 3,{\mathbb{R}})\). We are searching for a function u: \({\bar \Omega}\to {\mathbb{R}}^ 3\) such that \[ (1)\quad \Delta u=2H(u)u_{\alpha}\wedge u_{\beta}\quad on\quad \Omega \] satisfying \[ | u_{\alpha}|^ 2-| u_{\beta}|^ 2=u_{\alpha}\cdot u_{\beta}=0\quad on\quad \Omega, \] \(u|_{\partial\Omega}:\;\partial\Omega \to \Gamma\) is a parametrization of \(\Gamma.\) This is Plateau’s problem. The author investigates the behavior of sequences of solutions of (1), when H is not constant. He deals with a sequence of solutions \((u^ n)\) converging weakly in \(H^ 1(\Omega)\). The main result is the fact that, for a subsequence, one has the convergence in \(H^ 1_{loc}(\Omega \setminus S)\) where S is a subset of finite cardinal, included in \(\Omega\). The author proves that the weak limit is a solution of (1). Such a result has already been obtained by J. Sacks and K. Uhlenbeck [Ann. Math., II. Ser. 113, 1-24 (1981; Zbl 0462.58014)] for harmonic maps of \(S^ 2\) into compact Riemannian manifolds.
The author also investigates the behavior of the sequence near a point of S and shows that, at least, one solution of \(\Delta u=2H(u)u_{\alpha}\wedge u_{\beta}\) on \({\mathbb{R}}^ 2\) can be “found” at every point of S. Finally the paper studies solutions of (1) on \({\mathbb{R}}^ 2\) and shows that their energy \(\int_{{\mathbb{R}}^ 2}| \nabla u|^ 2\) is minored by a positive constant.
Reviewer: T.Rassias

MSC:
49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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