Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem | Zendy

Arshak Petrosyan | Zendy; Henrik Shahgholian | Zendy

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Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem

Author(s) -

Arshak Petrosyan,

Henrik Shahgholian

Publication year - 2007

Publication title -

american journal of mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 3.818

H-Index - 67

eISSN - 1080-6377

pISSN - 0002-9327

DOI - 10.1353/ajm.2007.0042

Subject(s) - obstacle problem , mathematics , obstacle , boundary (topology) , type (biology) , open set , connection (principal bundle) , sign (mathematics) , lipschitz continuity , continuation , mathematical analysis , potential theory , pure mathematics , geometry , computer science , law , ecology , political science , biology , programming language

We consider an obstacle-type problem Δu = f(x)χΩin D, u = |∇u| = 0on D \ Ω, where D is a given open set in ℝn and Ω is an unknown open subset of D. The problem originates in potential theory, in connection with harmonic continuation of potentials. The qualitative difference between this problem and the classical obstacle problem is that the solutions here are allowed to change sign. Using geometric and energetic criteria in delicate combination we show the C1,1 regularity of the solutions, and the regularity of the free boundary, below the Lipschitz threshold for the right-hand side.

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