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Almost minimizers for semilinear free boundary problems with variable coefficients
Author(s) -
de Queiroz Olivaine S.,
Tavares Leandro S.
Publication year - 2018
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/mana.201600103
Subject(s) - mathematics , boundary (topology) , exponent , matrix (chemical analysis) , mathematical analysis , variable (mathematics) , pure mathematics , minification , sense (electronics) , hölder condition , combinatorics , mathematical optimization , philosophy , linguistics , materials science , electrical engineering , composite material , engineering
We study regularity results for almost minimizers of the functionalF γ ( v ; Ω ) = ∫ Ω⟨ A ( x ) ∇ v , ∇ v ⟩ + q + ( x )( v + ) γ + q − ( x )( v − ) γ d x ,0 ≤ γ ≤ 1 ,where A is a matrix with Hölder continuous coefficients. In the case 0 < γ ≤ 1 we show that an almost minimizer belongs to C 1 , η , where the exponent η is related with the competition between the Hölder continuity of the matrix A , the parameter of almost minimization and γ. In some sense, this regularity is optimal. As far as the case γ = 0 is concerned, our results show that an almost minimizer is C 1 , θlocally in each phase { u > 0 } and { u < 0 } , improving in some sense a recent result of David & Toro.