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 language | English (US) |
|---|---|
| Pages (from-to) | 397-411 |
| Number of pages | 15 |
| Journal | Calculus of Variations and Partial Differential Equations |
| Volume | 41 |
| Issue number | 3-4 |
| DOIs | |
| State | Published - Oct 16 2010 |
| Externally published | Yes |
Bibliographical note
KAUST Repository Item: Exported on 2020-10-01Acknowledged 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.