The regularity of minimal surfaces defined over slit domains

David Samuel Kinderlehrer

Vol. 37 (1971), No. 1, 109–117

DOI: 10.2140/pjm.1971.37.109

Abstract

Let Ω denote the disc x₁² + x₂² < r² in the x = (x₁,x₂) plane from which the segment {0 ≦ x₁ < r,x₂ = 0} has been deleted. Suppose that u(x) ∈ C⁰(Ω) is a solution to the minimal surface equation in Ω ((1) below) and attains boundary values f(x₁) ∈ C^1,α(0 < α < 1) on the slit {0 ≦ x₁ < r,x₂ = 0}. We shall prove here that the gradient of u,Du = (u_x1,u_x2), is continuous at the origin x = 0.

Mathematical Subject Classification 2000

Primary: 53A10

Milestones

Received: 26 February 1970

Published: 1 April 1971