FG Analysis
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Spektraltheorie selbstadjungierter Operatoren im Hilbertraum und elliptischer Differentialoperatoren
(2019)
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.
Surfaces of prescribed mean curvature H(x,y,z) with one-to-one central projection onto a plane
(2016)
Minimal Surfaces
(2010)