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A boundary value problem in the hyperbolic space. (English) Zbl 0987.35058

Let \(M\) be the open unit ball in \(\mathbb{R}^3\) of center 0 and let \(g_{ij}(x)= {4\delta_{ij} \over\bigl(1- |x|^2\bigr)^2}\) be the hyperbolic metric on \(M\). Let \(\Omega\subset \mathbb{R}^2\) be a bounded domain with smooth boundary \(\partial\Omega\in C^{1,0}\), and let \((u,v)\) be the variables in \(\mathbb{R}^2\). The authors deal with the Dirichlet problem for a function \(Y: \overline \Omega\to M\) which satisfies the equation of prescribed mean curvature \[ \begin{cases} (1) \quad \nabla_{Y_u}Y_u+ \nabla_{Y_v}Y_v= -2H(Y)\cdot Y_u\wedge Y_v \text{ in } \Omega,\\ (2) \quad Y=g\text{ on }\partial \Omega,\end{cases} \] where \(H:M \to\mathbb{R}\) is a given continuous function, and \(g\in W^{2,p}(\Omega,\mathbb{R}^3)\) for \(1<p< \infty\), with \(\|g\|_\infty <1\). The authors prove solutions in a Sobolev space and present some regularity result of solutions.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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