Surfaces of prescribed mean curvature $H(x,y,z)$ with one-to-one central projection onto a plane

Friedrich Sauvigny

Vol. 281 (2016), No. 2, 481–509

DOI: 10.2140/pjm.2016.281.481

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; in the case $H=0$ this PDE was invented by T. Radó. 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. We also provide a Bernstein-type result for the case $H=0$ and classify the entire solutions of the minimal surface equation in central projection. We also solve the Dirichlet problem for $H=0$ by a variational method. We solve the Dirichlet problem for nonvanishing $H$ with compact support via a nonlinear continuity method, and we construct large $H$-surfaces bounding extreme contours by an approximation. Finally, we solve the Dirichlet problem on discs for the nonparametric $H$-surface equation in central projection under certain restrictions for the mean curvature.

Dedicated to the memory of Professor Stefan Hildebrandt in gratitude

Keywords

surfaces of prescribed mean curvature, solution of the Dirichlet problem, $H\mskip-2mu$-surface equation in central projection, large $H\mskip-2mu$-surfaces for Plateau's problem

Mathematical Subject Classification 2010

Primary: 35J60, 53A10

Milestones

Received: 6 August 2015

Revised: 11 August 2015

Accepted: 13 August 2015

Published: 16 February 2016