Hardy-type inequalities for vector fields with vanishing tangential components

Abstract

This note studies the Hardy-type inequalities for vector fields with the L 1 norm of the curl. In contrast to the well-known results in the whole space for the divergence-free vectors, we generalize the Hardy-type inequalities to the bounded domains and to the non-divergencefree vector fields with the tangential components on the boundary vanishing.

Existence and Regularity of Weak Solutions for a Thermoelectric Model

“…Next, we show H ∈ L p (Ω, R 3 ). We prove this by a duality method, which has been used in the proof of [26,Lemma 3.1]. Given F ∈ C ∞ c (Ω, R 3 ) and 1 < r < ∞, by Helmholtz-Weyl decomposition (see [2,Theorem 6.1] or [17,Theorem 2.1]), there exist w ∈ W 1,r (Ω, R 3 ) with ν × w = 0 on ∂Ω, χ ∈ W 1,r (Ω), and z ∈ H N (Ω) such that…”

Fractional integral operator for L1 vector fields and its applications

“…Another application is to establish some new weighted div-curl inequalities. By the way, we point out that div-curl inequalities involving L 1 norm have been studied by [2], [10], [14], [16], [11], [17], and the references therein.…”