2020/02/19 by Giao Bui, Bui, Giao, Fernando López‐García +3
Computer Science · Engineering · Mathematics · #76D07 #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in engineering #Primary: 26D15 #Secondary: 46E35
paper · pdf · doi:10.48550/arxiv.2002.07939
openalex publication_date 2020/02/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In a note published in 1925, G. H. Hardy stated the inequality ∑n=1^∞ ((1)/(n)∑k=1n ak )p ≤ ((p)/(p-1))p ∑n=1^∞ anp, for any non-negative sequence \an\n ≥ 1, and p>1. This inequality is known in the literature as the classical discrete Hardy inequality. It has been widely studied and several applications and new versions have been shown. In this work, we use a characterization of a weighted version of this inequality to exhibit a sufficient condition for the existence of solutions of the differential equation \rm div \bf u=f in weighted Sobolev spaces over a certain plane irregular domain. The solvability of this equation is fundamental for the analysis of the Stokes equations. The proof follows from a local-to-global argument based on a certain decomposition of functions which is also of interest for its applications to other inequalities or related results in Sobolev spaces, such as the Korn inequality.