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The Dirichlet problem for harmonic maps from the disk into the Euclidean n-sphere. (English) Zbl 0597.35022
Let $$\Omega =\{(x,y)\in {\mathbb{R}}^ 2:$$ $$x^ 2+y^ 2<1\}$$, $$S^ n=\{v\in {\mathbb{R}}^{n+1}:$$ $$| v| =1\}$$, $$n\geq 2$$, $$\gamma \in C^{2,\delta}(\partial \Omega,S^ n)$$ a nonconstant function $$(0<\delta <1)$$, $$\Sigma_ p=\{\sigma \in C^ 0(S^{n- 2},W_{\gamma}^{1,p}(\Omega,S^ n)),$$ $$\sigma$$ is not homotopic to a constant$$\}$$ where $$p>2$$, $$W_{\gamma}^{1,p}(\Omega,S^ n)=\{u\in W^{1,p}(\Omega,S^ n):\quad u=\gamma \quad on\quad \partial \Omega \},$$ $$\Sigma =\cup_{p>2}\Sigma_ p$$ and $$c=\inf_{\sigma \in \Sigma}(\max_{s\in S^{n-2}}E(\sigma (s))$$, where $$E(u)=\int_{\Omega}| \nabla u|^ 2 dx$$. Then:
Theorem. There exists at least $$u\in C^{2,\delta}({\bar \Omega},S^ n)$$ such that $$E(u)=c$$, $$-\Delta u=u| \nabla u|^ 2$$, $$u|_{\partial \Omega}=\gamma$$. Moreover if $$c=m$$ there exist infinitely many u when $$n\geq 3$$ (and at least two when $$n=2)$$, where $$m=\inf \{E(u):\quad u\in H^ 1(\Omega,{\mathbb{R}}^{n+1}),\quad u|_{\partial \Omega}=\gamma,\quad | u| =1\quad a.e.\}.$$
Reviewer: G.Bottaro

MSC:
 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions 58E12 Variational problems concerning minimal surfaces (problems in two independent variables) 35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:
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