Boundary Hardy inequality on functions of bounded variation

2024/05/16 by Adimurthi, Prosenjit Roy, Roy, Prosenjit +2
Mathematics · #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2405.09823

Abstract

Classical boundary Hardy inequality, that goes back to 1988, states that if 1 < p < ∞, ~Ω is bounded Lipschitz domain, then for all u ∈ C^∞_c(Ω), ∫_Ω \frac|u(x)|^pδ^p_Ω(x) dx ≤ C∫_Ω |∇ u(x) |^pdx, where δ_Ω(x) is the distance function from Ω^c. In this article, we address the long standing open question on the case p=1 by establishing appropriate boundary Hardy inequalities in the space of functions of bounded variation. We first establish appropriate inequalities on fractional Sobolev spaces W^s,1(Ω) and then Brezis, Bourgain and Mironescu's result on limiting behavior of fractional Sobolev spaces as s→ 1^- plays an important role in the proof. Moreover, we also derive an infinite series Hardy inequality for the case p=1.

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