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Journal Articles Mathematische Zeitschrift Year : 2013

A limiting case for the divergence equation

Abstract

We consider the equation $\text{div }{\mathbb Y}=f$, with $f$ a zero average function on the torus ${\mathbb T}^d$. In their seminal paper (J. Amer. Math. Soc. 2003), Bourgain and Brezis proved the existence of a solution ${\mathbb Y}\in W^{1,d}\cap L^\infty$ for a datum $f\in L^d$. We extend their result to the critical Sobolev spaces $W^{s,p}$ with $(s+1)p=d$ and $p\ge 2$. More generally, we prove a similar result in the scale of Triebel-Lizorkin spaces. We also consider the equation $\text{div }{\mathbb Y}=f$ in a bounded domain $\Omega$ subject to zero Dirichlet boundary condition.
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Dates and versions

hal-00639326 , version 1 (08-11-2011)
hal-00639326 , version 2 (09-11-2011)

Identifiers

  • HAL Id : hal-00639326 , version 2

Cite

Pierre Bousquet, Petru Mironescu, Emmanuel Russ. A limiting case for the divergence equation. Mathematische Zeitschrift, 2013, 274 (1-2), pp.427-460. ⟨hal-00639326v2⟩
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