Poincaré Inequalities in Punctured Domains

The classic Poincaré inequality bounds the L-norm of a function, f , orthogonal to a given function g in a domain Ω, in terms of some L-norm of its gradient in Ω. Suppose we now remove a set Γ from Ω and concentrate our attention on Λ = Ω \ Γ. This new domain might not even be connected and hence no Poincaré inequality can generally hold for it. This is so even if the volume of Γ is arbitrarily small. A Poincaré inequality does hold, however, if one makes the additional assumption that f has a finite L gradient norm on the whole of Ω, not just on Λ. The important point is that the Poincaré inequality thus obtained bounds the L-norm of f in terms of the L gradient norm on Λ (not Ω) plus an additional term that goes to zero as the volume of Γ goes to zero. This error term depends on Γ only through its volume. Another direction in which we generalize the Poincaré inequality is to the operator ∇+ iA(x) in place of the usual ∇. (Here, A is a given vector field.) Along the way we present some conjectures and open problems.