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Two-phase semilinear free boundary problem with a degenerate phase

  • Norayr Matevosyan
  • Arshak Petrosyan
Article

Abstract

We study minimizers of the energy functional
$$\int\limits_{D} [|\nabla u|^2+ \lambda(u^+)^p]\,{\rm d}x$$
for \({p\in (0,1)}\) without any sign restriction on the function u. The distinguished feature of the problem is the lack of nondegeneracy in the negative phase. The main result states that in dimension two the free boundaries \({\Gamma^+=\partial\{u>0\}\cap D}\) and \({\Gamma^-=\partial\{u<0\}\cap D}\) are C 1,α -regular, provided \({1-\epsilon_0<p<1}\) . The proof is obtained by a careful iteration of the Harnack inequality to obtain a nontrivial growth estimate in the negative phase, compensating for the apriori unknown nondegeneracy.

Mathematics Subject Classification (2000)

Primary 35R35 

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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Applied Mathematics and Theoretical PhysicsUniversity of CambridgeCambridgeUK
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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