Elcrat, Alan R.; Lancaster, Kirk E. Bernstein functions and the Dirichlet problem. (English) Zbl 0699.35095 SIAM J. Math. Anal. 20, No. 5, 1055-1068 (1989). The minimal surface problem arising from an associated Dirichlet problem is considered, where the nonconvex, planar domain D is given by a symmetric quadrilateral with re-entrant corner and the boundary function is continuous an \(\partial D\). The solution of this problem can be described in terms of the Weierstraß representation and the stereographic projection of its Gauss map. If the quadrilateral is truncated by replacing the re-entrant corner by a concave arc, a Bernstein function arises which has the same Gauss map image. This leads to a Riemann-Hilbert problem for the analytic function in the Weierstraß representation. This problem can be solved and leads to the existence of the surface. Reviewer: R.Heersink Cited in 4 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature 35Q15 Riemann-Hilbert problems in context of PDEs Keywords:minimal surface; symmetric quadrilateral; Weierstraß representation; stereographic projection; Gauss map; Bernstein function; Riemann-Hilbert problem; existence Software:kirch PDFBibTeX XMLCite \textit{A. R. Elcrat} and \textit{K. E. Lancaster}, SIAM J. Math. Anal. 20, No. 5, 1055--1068 (1989; Zbl 0699.35095) Full Text: DOI