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Differentiability of
minimal surfaces at the boundary
Frank David Lesley |
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Vol. 37 (1971), No. 1, 123–139
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Abstract |
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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,α, 0 < α < 1, then λ,μ,ν ∈ C1,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 ∈ 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
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Milestones
Received: 18 May 1970
Published: 1 April 1971
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Authors |
| Frank David Lesley | |
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