Multipolar Hardy inequalities and mutual interaction of the poles

2023/02/07 by Canale, Anna
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2302.03635

Abstract

In this paper we state the weighted Hardy inequality c∫_\mathbb R^N∑_i=1ⁿ (φ² )/(|x-a_i|²) μ(x)dx≤ ∫_\mathbb R^N |∇φ|² μ(x)dx +k ∫_ℝ^Nφ² μ(x)dx for any φ in a weighted Sobolev spaces, with c∈]0,c_o[ where c_o=c_o(N,μ) is the optimal constant, a₁,…,a_n∈ ℝ^N, k is a constant depending on μ. We show the relation between c and the closeness to the single pole. To this aim we analyze in detail the difficulties to be overcome to get the inequality.

Related

• Multipolar potentials and weighted Hardy inequalities
• Weighted multipolar Hardy inequalities and evolution problems with Kolmogorov operators perturbed by singular potentials
• Improved Hardy inequalities with a class of weights
• Weighted Hardy's inequalities and Kolmogorov-type operators
• A class of weighted Hardy inequalities and applications to evolution\n problems