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The Dirichlet problem for the minimal surfaces equation and the plateau problem at infinity. (English) Zbl 1063.53007

Let \(M\) be a complete minimal surface in \(\mathbb{R}^{3}\) with finite total curvature and embedded ends. Each end is asymptotic to a plane or to a half-plane and we can associate to each end a vector, which is called the flux vector of the end. The sum of the flux vectors over all ends is zero. An interesting and still open problem concerning minimal surfaces is the Plateau problem at infinity: Given a finite number of vectors such that their sum is zero, can we find a minimal surface which has these vectors as flux vectors?
C. Cosin and A. Ros in [Indiana Univ. Math. J. 50, 847–879 (2001; Zbl 1041.53004)] gave a description of the space of solutions of the Plateau problem at infinity with an asymptotic behaviour which is symmetric with respect to a horizontal plane (i.e. all the flux vectors are horizontal). Here the author gives a more constructive proof of the result of Cosin and Ros. To this purpose the author studies, firstly, the Dirichlet problem for the minimal surface equation, for unbounded domains and proves some results about the boundary behaviour of solutions of the Dirichlet problem.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature

Citations:

Zbl 1041.53004
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