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

2022/02/24 by Ujjal Das, Das, Ujjal, Yehuda Pinchover +1 · 3 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2202.12324

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 given 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] \rm dx is nonnegative in 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.

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• Optimal Hardy-weights for the Finsler p-Dirichlet integral with a potential
• L^p Hardy inequalities with homogeneous weights
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