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Plateau's problem for parametric double integrals: I. Existence and regularity in the interior
Author(s) -
Hildebrandt Stefan,
von der Mosel Heiko
Publication year - 2003
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.10080
Subject(s) - mathematics , pure mathematics , mathematical analysis , parametric statistics , plateau (mathematics) , boundary (topology) , domain (mathematical analysis) , function (biology) , multiple integral , zero (linguistics) , class (philosophy) , linguistics , statistics , philosophy , evolutionary biology , artificial intelligence , computer science , biology
We study Plateau's problem for two‐dimensional parametric integrals$${\cal F}(X) := \int_{B} F(X, X_{u} \wedge X_{v}) \,du \,dv,$$ the Lagrangian F ( x, z ) of which is positive definite and at least semi‐elliptic. It turns out that there always exists a conformally para‐me‐trized minimizer. Any such minimizer X is seen to be Hölder‐continuous in the parameter domain B and continuous up to its boundary. If F possesses a perfect dominance function G of class C 2 , we can establish higher regularity of X in the interior. In fact, we prove X ⊆ H loc 2,2 ( B , ℝ n ) ∩ C 1,σ ( B , ℝ n ) for some σ > 0. Finally, we discuss the existence of perfect dominance functions.