The Dirichlet problem for the minimal surface equation on unbounded planar domains. (English) Zbl 0696.49069

We prove existence and regularity results depending on the boundary data f (or g) for solutions u (or v) of the minimal surface equation on a certain unbounded domain \(\Omega\). Suppose \(\Omega\) is convex and contained in a band or proper sector, if f, g are uniformly continuous and bounded on \(\partial \Omega\) then so is the solution u, v, respect. This also holds for “convex city maps”. If f-g\(\leq A\) on \(\partial \Omega\) then u-v\(\leq A\) on \(\Omega\). Our techniques can be generalized to \(\Omega \subset R^ n\) (this is part of a joint work appearing in Proc. in Honor to M. do Carmo). Main generalizations for the mean curvature equation has been done by Collin and R. Krust (to appear in Bull. Soc. Math. Fr.), and by J.-F. Hwang [Ann. Sc. Norm. Super Pisa, Cl. Sci., IV. Ser. 15, No.3, 341-355 (1988)].
Reviewer: R.Sa Earp


49Q05 Minimal surfaces and optimization
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature