Given any Γ=γ(S^1)⊂R^2, image of a Lipschitz curve γ:S^1→R^2, not necessarily injective, we provide an explicit formula for computing the value of (Formula presented.) where the infimum is computed among all Lipschitz maps u:B_1(0)→R^2 having boundary datum γ. This coincides with the area of a minimal disk spanning Γ, i.e., a solution of the Plateau problem of disk type. The novelty of the results relies in the fact that we do not assume the curve γ to be injective and our formula allows arbitrary self-intersections.
On the singular planar Plateau problem / Caroccia Marco; Scala Riccardo. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - ELETTRONICO. - 63:(2024), pp. 242.1-242.32. [10.1007/s00526-024-02853-y]
On the singular planar Plateau problem
Caroccia Marco;
2024
Abstract
Given any Γ=γ(S^1)⊂R^2, image of a Lipschitz curve γ:S^1→R^2, not necessarily injective, we provide an explicit formula for computing the value of (Formula presented.) where the infimum is computed among all Lipschitz maps u:B_1(0)→R^2 having boundary datum γ. This coincides with the area of a minimal disk spanning Γ, i.e., a solution of the Plateau problem of disk type. The novelty of the results relies in the fact that we do not assume the curve γ to be injective and our formula allows arbitrary self-intersections.
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