Hardy-type inequality in variable exponent Lebesgue spaces derived from nonlinear problem

2014/07/23 by Sylwia Dudek, Dudek, Sylwia, Iwona Skrzypczak +1
Mathematics · #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1407.6226

openalex publication_date 2014/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We derive a family of weighted Hardy-type inequalities in the variable exponent Lebesgue space with an additional term of the form ∫_Ω |ξ|^p(x) μ_1,β(dx)\leqslant ∫_Ω|∇ ξ|^p(x)μ_2,β(dx)+∫_Ω|ξlog ξ |^p(x) μ_3,β(dx), where ξ is any compactly supported Lipschitz function. The involved measures depend on a certain solution to the partial differential inequality involving p(x)-Laplacian -Δ_p(x) u\geqslant Φ, where Φ is a given locally integrable function, and u is defined on an open and not necessarily bounded subset Ω⊆ℝⁿ , and a certain parameter β. We derive new Caccioppoli-type inequality for the solution u. As its consequence we get Hardy-type inequality. We illustrate the result by several one-dimensional examples.

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