On existence of minimizers for weighted L^p-Hardy inequalities on C^1,γ-domains with compact boundary

2023/03/06 by Das, Ujjal, Pinchover, Yehuda, Devyver, Baptiste
#35B09 #35J20 #35J92 #49J40 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2303.03527

Abstract

Let p ∈ (1,∞), α∈ ℝ, and Ω\subsetneq ℝ^N be a C^1,γ-domain with a compact boundary ∂ Ω, where γ∈ (0,1]. Denote by δ_Ω(x) the distance of a point x∈ Ω to ∂ Ω. Let \widetildeW^1,p;α₀(Ω) be the closure of C_c^∞(Ω) in \widetildeW^1,p;α(Ω), where \widetildeW^1,p;α(Ω):= \φ∈ W^1,p_loc (Ω) | ( ‖ |∇ φ |‖_{L^p(Ω;δΩ^-α)}^p + ‖φ‖_{L^p(Ω;δΩ^-(α+p))}^p)lt;∞ \. We study the following two variational constants: the weighted Hardy constant H_α,p(Ω): = inf \∫_Ω |∇ φ|^p δ_Ω^-α dx \biggm| ∫_Ω |φ|^p δ_Ω^-(α+p) dx = 1, φ∈ \widetildeW^1,p;α₀(Ω) \ , and the weighted Hardy constant at infinity λ_α,p^∞(Ω) :=sup_{K\Subset Ω} inf_W^1,p_c(Ω∖ K) \∫_Ω∖ K |∇ φ|^p δ_Ω^-α dx \biggm| ∫_Ω∖ K |φ|^p δ_Ω^-(α+p) dx=1 \. We show that H_α,p(Ω) is attained if and only if the spectral gap Γ_α,p(Ω):= λ_α,p^∞(Ω)-H_α,p(Ω) is strictly positive. Moreover, we obtain tight decay estimates for the corresponding minimizers.

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