Abstract
When we consider surfaces of prescribed mean curvature H with a one-to-one orthogonal projection onto a plane, we have to study the nonparametric H-surface equation. Now the H-surfaces with a one-to-one central projection onto a plane lead to an interesting elliptic differential equation, which has been discovered for the case H = 0 already by T. Radó in 1932. We establish the uniqueness of the Dirichlet problem for this H-surface equation in central projection and develop an estimate for the maximal deviation of large H-surfaces from their boundary values, resembling an inequality by J. Serrin from 1969.
We solve the Dirichlet problem for nonvanishing H with compact support via a nonlinear continuity method. Here we introduce conformal parameters into the surface and study the well-known H-surface system. Then we combine these investigations with a differential equation for its unit normal, which has been developed by the author for variable H in 1982. Furthermore, we construct large H-surfaces bounding extreme contours by an approximation.
Here we only provide an overview on the relevant proofs; for the more detailed derivations of our results, we refer the readers to the author’s investigations in the Pacific Journal of Mathematics and the Milan Journal of Mathematics.
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Dedicated to Professor Dr. Robert Finn and Professor Dr. Vsevolod Solonnikov in very high respect, remembering a wonderful conference from the 21st to the 25th of August 2017 at the University of Vilnius
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Sauvigny, F. Solution of boundary value problems for surfaces of prescribed mean curvature H (x, y, z) with 1–1 central projection via the continuity method. Lith Math J 58, 320–328 (2018). https://doi.org/10.1007/s10986-018-9399-y
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DOI: https://doi.org/10.1007/s10986-018-9399-y