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- W2899018341 abstract "We consider critical points of the geometric obstacle problem on vectorial maps u:B2⊂R2→RN∫B2|∇u|2subject to u∈RNBN(0). Our main result is C1,α-regularity for any α<1. Technically, we split the map u=λv, where v:B2→SN−1 is the vectorial component and λ=|u| the scalar component measuring the distance to the origin. While v satisfies a weighted harmonic map equation with weight λ2, λ solves the obstacle problem for∫B2|∇λ|2+λ2|∇v|2,subject to λ≥1, where |∇v|2∈L1(B2). We then play ping-pong between the increases in the regularity of λ and v to obtain finally the C1,α-result. On considère des points critiques du problème de l'obstacle géométrique pour des applications vectorielles u:B2⊂R2→RN∫B2|∇u|2avec u∈RNBN(0). Notre résultat principal est la régularité C1,α pour tout α<1 de ces points critiques. Du point de vue technique, on décompose l'application u=λv, où v:B2→SN−1 est la composante vectorielle et λ=|u| la composante scalaire qui mesure la distance à l'origine. Alors que v satisfait une équation de type applications harmoniques avec un poids λ2, la fonction λ, elle, est solution du problème de l'obstacle pour∫B2|∇λ|2+λ2|∇v|2,avec λ≥1, où |∇v|2∈L1(B2). On joue alors au ping-pong en gagnant tour à tour de la régularité pour λ puis v pour obtenir finalement le résultat de régularité C1,α." @default.
- W2899018341 created "2018-11-09" @default.
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- W2899018341 date "2020-04-01" @default.
- W2899018341 modified "2023-09-24" @default.
- W2899018341 title "On C1,-regularity for critical points of a geometric obstacle-type problem" @default.
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- W2899018341 doi "https://doi.org/10.1016/j.matpur.2020.01.004" @default.
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