The regularity of minimal surfaces defined over slit domains

Abstract

Let Ω denote the disc x\ + x\ < r 2 in the x = (xi, x 2 ) plane from which the segment {0 ^ Xi < r, x 2 -0} has been deleted.Suppose that u{x) e C° (Ω) is a solution to the minimal surface equation in Ω((l) below) and attains boundary values /(a?i) e C Utt (0 < a <1) on the slit {0 g x t < r, x 2 = 0}.We shall prove here that the gradient of u, Du = (u Xl ,u X2 ), is continuous at the origin x = 0.veSf JϊJ Evidently, a solution to A, if it exists, satisfies (1) in the set 109

Locations

• Pacific Journal of Mathematics - View - PDF
• Project Euclid (Cornell University) - View - PDF