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A limiting case for the divergence equation |
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| Authors: | Pierre Bousquet Petru Mironescu Emmanuel Russ |
| Affiliation: | 1. LATP, Centre de Mathématiques et Informatique, Aix-Marseille Université, CNRS, UMR 7353, 39 rue Frédéric Joliot-Curie, 13453, Marseille Cedex 13, France 2. Université de Lyon, Université Lyon 1, 69622, Villeurbanne-Cedex, France 3. Institut Camille Jordan, CNRS, UMR 5208, 43 blvd du 11 novembre 1918, 69622, Villeurbanne-Cedex, France 4. Institut Fourier, Université Joseph Fourier; CNRS, UMR 5582, 100 rue des maths, BP 74, BP 74, 38402, Saint-Martin d’Hères Cedex, France |
| 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, Bourgain and Brezis [J Am Math Soc 16(2):393–426, 2003 (electronic)] proved the existence of a solution $mathbb{Y }in W^{1,d}cap L^infty $ for a datum $fin L^d$ . We extend their result to the critical Sobolev spaces $W^{s,p}$ with $(s+1)p=d$ and $pge 2$ . 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 $varOmega $ subject to zero Dirichlet boundary condition. |
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