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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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