Type: Article
Publication Date: 1971-04-01
Citations: 37
DOI: https://doi.org/10.2140/pjm.1971.37.123
Let Γ be a Jordan curve in R z and F(z) = (u(z) 9 v(z), w(z)): {\z\ ^ 1} -» R B 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 ΓeC ί>a , 0 < a < 1, then ;,/ί,y6(?β for \z\^l and if Γe 11 then λ r , μ', v !have modulus of continuity Kt log 1/ί for | z | ^ 1; K and the Holder constants depend only on the geometry of Γ. 2. If ΓeC n ω{t) , n^2, where ω(t) is a modulus of continuity satisfying a Dini condition, then λ, μ, v e C n > ω * (ί) for | z \ ^ 1, where ω*(t) is a certain modulus of continuity.Once again ω* depends only on Γ.