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Obstacle Problems with Linear Growth: Hölder Regularity for the Dual Solution
Author(s) -
Bildhauer Michael,
Fuchs Martin
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200112)232:1<5::aid-mana5>3.0.co;2-k
Subject(s) - mathematics , smoothness , duality (order theory) , sobolev space , limit of a sequence , dual (grammatical number) , limit (mathematics) , sequence (biology) , trace (psycholinguistics) , regular polygon , obstacle problem , pure mathematics , zero (linguistics) , class (philosophy) , mathematical analysis , obstacle , function (biology) , combinatorics , variational inequality , geometry , art , linguistics , philosophy , literature , artificial intelligence , evolutionary biology , biology , computer science , genetics , political science , law
For a strictly convex integrand f : ℝ n → ℝ with linear growth we discuss the variational problem$$J(u) = \int_{\Omega} f (\nabla u) dx\ \longrightarrow\ {\rm min}$$among mappings u : ℝ n ⊃ Ω → ℝ of Sobolev class W 1 1 with zero trace satisfying in addition u ≥ ψ for a given function ψ such that ψ| ∂Ω < 0. We introduce a natural dual problem which admits a unique maximizer σ. In further sections the smoothness of σ is investigated using a special J ‐minimizing sequence with limit u * ∈ C 1,α (Ω) for which the duality relation $\sigma = {^{\partial f} \over _{\partial P}} (\nabla u^{\ast})$ holds.