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.
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