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On solutions of the exterior Dirichlet problem for the minimal surface equation. (English) Zbl 0820.35038

Summary: Uniqueness and existence results for boundary value problems for the minimal surface equation \(\text{div} (\nabla u/ \sqrt {1 + | \nabla u |^ 2}) = 0\) on exterior domains obtained by Langévin-Rosenberg and Krust in dimension two are generalized to arbitrary dimensions. A suitable \(n\)-dimensional version of the maximum principle at infinity is given.

MSC:

35J25 Boundary value problems for second-order elliptic equations
35B50 Maximum principles in context of PDEs
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References:

[1] Giusti, E., Minimal Surfaces and Functions of Bounded Variation (1984), Birkhäuser Verlag: Birkhäuser Verlag Boston, Basel, Stuttgart · Zbl 0545.49018
[2] Haar, A., Über das Plateausche Problem, Math. Ann., Vol. 97, 124-258 (1927) · JFM 52.0710.02
[3] Jenkins, H.; Serrin, J., The Dirichlet Problem for the Minimal Surface Equation in Higher Dimension, J. Reine Ang. Math., Vol. 229, 170-187 (1968) · Zbl 0159.40204
[4] Korevaar, N.; Kusner, R.; Solomon, B., The Structure of Complete Embedded Surfaces with Constant Mean Curvature, J. Differential Geometry, Vol. 30, 465-503 (1989) · Zbl 0726.53007
[5] Krust, R., Remarques sur le problème extérieur de Plateau, Duke Math. J., Vol. 59, 161-173 (1989) · Zbl 0709.49022
[6] Kuwert, E., Embedded Solutions for Exterior Minimal Surface Problems, Manuscripta math., Vol. 70, 51-65 (1990) · Zbl 0717.49034
[7] Langévin, R.; Rosenberg, H., A Maximum Principle at Infinity for Minimal Surfaces and Applications, Duke Math. J., Vol. 57, 819-828 (1988) · Zbl 0667.49024
[9] Meeks, W. H.; Yau, S. T., The Existence of Embedded Minimal Surfaces and the Problem of Uniqueness, Math. Z., Vol. 179, 151-168 (1982) · Zbl 0479.49026
[10] Moser, J., On Harnack’s Theorem for Elliptic Differential Equations, Comm. Pure Appl. Math., Vol. 14, 577-591 (1961) · Zbl 0111.09302
[11] Osserman, R., A Survey of Minimal Surfaces (1986), Dover publ: Dover publ New York · Zbl 0209.52901
[12] Schoen, R., Uniqueness, Symmetry and Embeddedness of Minimal Surfaces, J. Differential Geometry, Vol. 18, 791-809 (1983) · Zbl 0575.53037
[13] Tomi, F.; Ye, R., The Exterior Plateau Problem, Math. Z., Vol. 205, 223-245 (1990) · Zbl 0719.53006
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