Differentiability of minimal surfaces at the boundary

Frank David Lesley

Vol. 37 (1971), No. 1, 123–139

DOI: 10.2140/pjm.1971.37.123

Abstract

Let Γ be a Jordan curve in R³ and F(z) = (u(z),v(z),w(z)): {|z|≦ 1}→ R⁸ be a solution of Plateau’s problem for Γ, where z = x + iy are isothermal parameters. Then u,v,w are harmonic in {|z| < 1} and are the real parts of analytic functions λ,μ,v. Using the Poisson integral and the defining properties of minimal surfaces, Kellogg’s theorem for conformal mapping is generalized by proving: 1. If Γ ∈ C^1,α, 0 < α < 1, then λ,μ,ν ∈ C^1,α for |z|≦ 1 and if Γ ∈ 11 then λ′,μ′,ν′ have modulus of continuity Ktlog 1∕t for |z|≦ 1;K and the Holder constants depend only on the geometry of Γ. 2. If Γ ∈ C^n.ω(t),n ≧ 2, where ω(t) is a modulus of continuity satisfying a Dini condition, then λ,μ,ν ∈ C^n.ω∗(t) for |z|≦ 1, where ω^∗(t) is a certain modulus of continuity. Once again ω^∗ depends only on Γ.

Mathematical Subject Classification 2000

Primary: 53A10

Secondary: 49F10

Milestones

Received: 18 May 1970

Published: 1 April 1971