A minimum problem with free boundary in Orlicz spaces

We consider the optimization problem of minimizing ∫Ω G (| ∇ u |) + λ χ{u > 0} d x in the class of functions W1, G (Ω) with u - φ0 ∈ W01, G (Ω), for a given φ0 ≥ 0 and bounded. W1, G (Ω) is the class of weakly differentiable functions with ∫Ω G (| ∇ u |) d x < ∞. The conditions on the...

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Detalles Bibliográficos
Autores principales: Martínez, S., Wolanski, N.
Formato: Artículo publishedVersion
Publicado: 2008
Materias:
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00018708_v218_n6_p1914_Martinez
Aporte de:Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires Ver origen
Descripción
Sumario:We consider the optimization problem of minimizing ∫Ω G (| ∇ u |) + λ χ{u > 0} d x in the class of functions W1, G (Ω) with u - φ0 ∈ W01, G (Ω), for a given φ0 ≥ 0 and bounded. W1, G (Ω) is the class of weakly differentiable functions with ∫Ω G (| ∇ u |) d x < ∞. The conditions on the function G allow for a different behavior at 0 and at ∞. We prove that every solution u is locally Lipschitz continuous, that it is a solution to a free boundary problem and that the free boundary, Ω ∩ ∂ {u > 0}, is a regular surface. Also, we introduce the notion of weak solution to the free boundary problem solved by the minimizers and prove the Lipschitz regularity of the weak solutions and the C1, α regularity of their free boundaries near "flat" free boundary points. © 2008 Elsevier Inc. All rights reserved.