Radial limits of bounded nonparametric prescribed mean curvature surfaces

Abstract

Consider a solution f is an element of C-2 (Omega)of a prescribed mean curvature equation

div del f/root 1+vertical bar del f vertical bar(2) = 2H(x, f) in Omega,

where Omega subset of R-2 is a domain whose boundary has a corner at O = (0, 0) is an element of partial derivative Omega. If sup(x is an element of Omega) vertical bar f(x)vertical bar and sup(x is an element of Omega) vertical bar H (x, f(x))vertical bar are both finite and Omega has a reentrant corner at O, then the (nontangential) radial limits of integral at O,

R integral (theta) := lim(r down arrow 0) integral(r cos theta, r sin theta),

are shown to exist, independent of the boundary behavior of f on partial derivative Omega, and to have a specific type of behavior. If sup(x is an element of Omega) vertical bar f(x)vertical bar and sup(x is an element of Omega) vertical bar H (x, f(x))vertical bar are both finite and the trace of f on one side has a limit at O, then the (nontangential) radial limits of f at O exist, the tangential radial limit of f at O from one side exists and the radial limits have a specific type of behavior.