Vol. 37, No. 1, 1971

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Differentiability of minimal surfaces at the boundary

Frank David Lesley

Vol. 37 (1971), No. 1, 123–139
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 C1, 0 < α < 1, then λ,μ,ν C1 for |z|1 and if Γ 11 then λhave modulus of continuity Ktlog 1∕t for |z|1;K and the Holder constants depend only on the geometry of Γ. 2. If Γ Cn.ω(t),n 2, where ω(t) is a modulus of continuity satisfying a Dini condition, then λ,μ,ν Cn.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
Authors
Frank David Lesley