Abstract
Let x(w), w=u+iv ∈ B, be a minimal surface in ℝ3 which is bounded by a configuration 〈Γ, S〉 consisting of an arc Γ and of a surface S with boundary. Suppose also that x(w) is area minimizing with respect to 〈Γ, S〉. Under appropriate regularity assumptions on Γ and S, we can prove that the first derivatives of x(u, v) are Hölder continuous with the exponent α=1/2 up to the “free part” of ∂B which is mapped by x(w) into S. An example shows that this regularity result is optimal.
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Hildebrandt, S., Nitsche, J.C.C. Optimal boundary regularity for minimal surfaces with a free boundary. Manuscripta Math 33, 357–364 (1981). https://doi.org/10.1007/BF01798233
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DOI: https://doi.org/10.1007/BF01798233