Overdetermined problems in unbounded domains with Lipschitz singularities

Alberto Farina, Enrico Valdinoci

Research output: Contribution to journalArticle

13 Citations (Scopus)

Abstract

We study the overdetermined problem δu + f(u) = 0 in Ω, u = 0 on ∂Ω, ∂vu = c on Σ, where Ω is a locally Lipschitz epigraph, that is C3 on T Γ⊆ ∂Ω with ∂Ω \Σ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets fi for which monotone solutions exist. In particular, if l∈ℝn is a cone and either n = 2 or n = 3 and f≥0, then there exists no solution of δu +f(u) = 0 in l, u > 0 in l, u = 0 on ∂l, ∂vu = c on∂l/{0}. This answers a question raised by Juan Luis Vázquez.

Original languageEnglish
Pages (from-to)965-974
Number of pages10
JournalRevista Matematica Iberoamericana
Volume26
Issue number3
DOIs
Publication statusPublished - 1 Jan 2010
Externally publishedYes

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