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

Sá Earp, Ricardo; Rosenberg, Harold

J. Math. Pures Appl., IX. Sér. 68, No. 2, 163-183 (1989).

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

MSC:

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

Keywords:

modulus of continuity; Phragmèn Lindelöf techniques; regularity results; minimal surface equation; convex city maps; mean curvature equation