TY - JOUR
T1 - Overdetermined problems in unbounded domains with Lipschitz singularities
AU - Farina, Alberto
AU - Valdinoci, Enrico
PY - 2010/1/1
Y1 - 2010/1/1
N2 - 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.
AB - 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.
KW - Elliptic partial differential equations
KW - Nonexistence of solutions
KW - Rigidity results
UR - http://www.scopus.com/inward/record.url?scp=78649666070&partnerID=8YFLogxK
U2 - 10.4171/RMI/623
DO - 10.4171/RMI/623
M3 - Article
AN - SCOPUS:78649666070
SN - 0213-2230
VL - 26
SP - 965
EP - 974
JO - Revista Matematica Iberoamericana
JF - Revista Matematica Iberoamericana
IS - 3
ER -