Two-phase semilinear free boundary problem with a degenerate phase

Norayr Matevosyan, Arshak Petrosyan

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


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
Issue number3-4
StatePublished - Oct 16 2010
Externally publishedYes

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledged KAUST grant number(s): KUK-I1-007-43
Acknowledgements: We would like to thank the anonymous referee for valuable comments that have contributed to the improvement of the paper. N. Matevosyan is partially supported by the WWTF (Wiener Wissenschafts, Forschungs und Technologiefonds) project No. CI06 003 and by award No. KUK-I1-007-43, made by King Abdullah University of Science and Technology (KAUST). A. Petrosyan is supported in part by NSF grant DMS-0701015.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.


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