Hardy-Poincaré inequalities with boundary singularities

We are interested in variational problems involving weights that are singular at a point of the boundary of the domain. More precisely, we study a linear variational problem related to the Poincaré inequality and to the Hardy inequality for maps in H 0 ¹(Ω), where Ω is a bounded domain in ℝ ^N, N ≥ 2, with 0 ∈ ∂ Ω. In particular, we give sufficient and necessary conditions so that the best constant is achieved. © Copyright 2012 Royal Society of Edinburgh.