Overdetermined problems in unbounded domains with Lipschitz singularities

We study the overdetermined problem ⎧⎨ ⎩ Δu + f(u) = 0 in Ω, u = 0 on ∂Ω, ∂νu = c on Γ, where Ω is a locally Lipschitz epigraph, that is C3 on Γ ⊆ ∂Ω, with ∂Ω \ Γ consisting in nonaccumulating, countably many points. We provide a geometric inequality that allows us to deduce geometric properties of the sets Ω for which monotone solutions exist. In particular, if C ∈ 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 C , u > 0 in C , u = 0 on ∂C , ∂νu = c on ∂C \ {0}. This answers a question raised by Juan Luis Vázquez. Introduction Let n 2. We consider an epigraph in R, that is Ω := { (x′, xn) ∈ Rn−1 × R s.t. xn > Φ(x′) } . We suppose that Ω is locally Lipschitz and that it is C except, at most, at a countable family of points that do not accumulate. 2000 Mathematics Subject Classification: 35J25, 35J20, 35B65.