The space of Hardy-weights for quasilinear equations: Maz’ya-type characterization and sufficient conditions for existence of minimizers

14.12.2022, 13:30  –  Haus 9, Raum 0.17 und Zoom
Forschungsseminar Diskrete Spektraltheorie

Prof. Yehuda Pinchover (Technion)

Abstract: Let p ∈ (1,∞) and  Ω⊂ℝN be a domain. 

Let A:=(aij) ∈ Lloc(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.

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