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⊂ℝ^N be a domain. 

Let A:=(a_ij) ∈ L^∞_loc(Ω ; ℝ^{N× N}) be  a symmetric and locally uniformly positive definite matrix. Set |ξ|_A²:= ∑_i,j=1^N a_ij(x) ξ_i ξ_j,  ξ ∈ ℝ^N, and let V be a real valued potential in a certain local Morrey space. We assume that the energy functional

Q_p,A,V(ϕ) := ∫_Ω (|∇ ϕ|_A^p + V|ϕ|^p) dx 

is nonnegative on W^1,p(Ω)∩ C_c(Ω).
   
We introduce a generalized notion of Q_p,A,V-capacity and characterize  the space of all Hardy-weights for the functional Q_p,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.