A limiting case for the divergence equation

We consider the equation , with a zero average function on the torus . In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution for a datum . We extend their result to the critical Sobolev spaces with and . More generally, we pro...

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Bibliographic Details
Published inMathematische Zeitschrift Vol. 274; no. 1-2; pp. 427 - 460
Main Authors Bousquet, Pierre, Mironescu, Petru, Russ, Emmanuel
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.06.2013
Springer
Subjects
Analysis of PDEs
Mathematics
Mathematics and Statistics
46E35
Divergence equation
Critical embedding
42B05
35F05
Triebel–Lizorkin spaces
35F15
Triebel-Lizorkin spaces
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ISSN0025-5874
1432-1823
DOI10.1007/s00209-012-1077-x

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Summary:We consider the equation , with a zero average function on the torus . In their seminal paper, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution for a datum . We extend their result to the critical Sobolev spaces with and . More generally, we prove a similar result in the scale of Triebel–Lizorkin spaces. We also consider the equation in a bounded domain subject to zero Dirichlet boundary condition.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-012-1077-x