Two-phase semilinear free boundary problem with a degenerate phase

Norayr Matevosyan, Arshak Petrosyan

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

Abstract

We study minimizers of the energy functional ∫D[{pipe}∇u{pipe}2 + λ(u+)p]dx for p ∈ (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 Γ+ = ∂{u > 0} ∩ D andΓ- = ∂{u < 0} ∩ D are C1,α-regular, provided 1 - ∈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. © 2010 Springer-Verlag.
Original languageEnglish (US)
Pages (from-to)397-411
Number of pages15
JournalCalculus of Variations and Partial Differential Equations
Volume41
Issue number3-4
DOIs
StatePublished - Oct 16 2010
Externally publishedYes

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