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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. [ABSTRACT FROM AUTHOR]

Abstract: One admires rotational staircases in classical buildings since centuries. In particular, we are fascinated and inspired by the beautiful winding staircase (please, regard the picture below) in the center of the recently constructed University Library of the Brandenburgian Technical University at Cottbus by the bureau of architects Herzog & de Meuron from Basel. The sophisticated mathematician directly recognizes this staircase being a rotational minimal surface – namely the well-known helicoid – with a multiply covering projection onto the plane, solving a semi-free boundary value problem. We now ask the question, in which class of surfaces this helicoid is uniquely determined. Furthermore, we examine in how far the boundary values can be perturbed such that neighboring surfaces still exist. Both questions being affirmatively answered, we receive the stability of this boundary value problem. Finally, we investigate that our surface realizes a global minimum of area in the class of all parametric minimal surfaces solving an adequate mixed boundary value problem. Here we study one-to-one harmonic mappings onto the universal covering of the plane. This is achieved on the basis of our joint investigations with Professor Stefan Hildebrandt from the University of Bonn. Since H. Catalan was the first to classify the helicoid among ruled minimal surfaces and J. Plateau contributed, besides his inspiring experiments with soap bubbles, also his name to our central problem, I would like to present this treatise in the French language. During the construction of our University Library I got acquainted to the responsible architect for this project from the bureau Herzog & de Meuron, Frau Christine Binswanger and would like to dedicate this work to her with great respect. In her home city of Basel, classical Analysis could originally be developed by members of the Bernoulli family and Leonhard Euler. [ABSTRACT FROM AUTHOR]