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Article Dans Une Revue Indiana University Mathematics Journal Année : 2020

On the existence of vector fields with nonnegative divergence in rearrangement-invariant spaces

Résumé

We investigate the existence of solutions of \begin{equation} divF=\mu,\text{ on } \mathbf{R}^{d}. \tag{*} \end{equation} Here, $\mu\geq0$ is a Radon measure, and we look for a solution $F\in X(\mathbf{R}^{d}\rightarrow\mathbf{R}^{d})$, where $X$ is a rearrangement-invariant space. We first prove the equivalence of the following assertions: (i) (*) has a solution for some nontrivial $\mu$; (ii) the function $x\mapsto |x|^{1-d}1_{B^{c}}(x)$ belongs to $X$. Here, $B$ is the unit ball in $\mathbf{R}^{d}$. We next investigate the solvability of (*) when $\mu$ is fixed. A sufficient condition is that $I_{1}\mu\in X$, where $ I_{1}\mu$ is the 1-Riesz potential of $\mu$. This condition turns out to be also necessary when the Boyd indexes of $X$ belong to $(0,1)$. Our analysis generalizes the one of Phuc and Torres (2008) when $X=L^{p}$.
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Dates et versions

hal-01664360 , version 1 (14-12-2017)
hal-01664360 , version 2 (16-12-2018)

Identifiants

Citer

Eduard Valentin Curca. On the existence of vector fields with nonnegative divergence in rearrangement-invariant spaces. Indiana University Mathematics Journal, 2020, 69 (1), pp.119-136. ⟨10.1512/iumj.2020.69.7950⟩. ⟨hal-01664360v2⟩
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