Prof. Yehuda Pinchover (Technion)
Abstract: Let p ∈ (1,∞) and Ω⊂ℝN be a domain.
Let A:=(aij) ∈ L∞loc(Ω ; ℝN× N) be a symmetric and locally uniformly positive definite matrix. Set |ξ|A2:= ∑i,j=1N aij(x) ξi ξj, ξ ∈ ℝN, and let V be a real valued potential in a certain local Morrey space. We assume that the energy functional
Qp,A,V(ϕ) := ∫Ω (|∇ ϕ|Ap + V|ϕ|p) dx
is nonnegative on W1,p(Ω)∩ Cc(Ω).
We introduce a generalized notion of Qp,A,V-capacity and characterize the space of all Hardy-weights for the functional Qp,A,V, extending Maz'ya's well known characterization of the space of Hardy-weights for the p-Laplacian. In addition, we provide various sufficient conditions on the potential V and the Hardy-weight g such that the best constant of the corresponding variational problem is attained in an appropriate Beppo Levi space.
This talk is based on a joint work with Ujjal Das.
