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On the equation $\operatorname{div}Y=f$ and application to control of phases

Authors: Jean Bourgain and Haïm Brezis
Journal: J. Amer. Math. Soc. 16 (2003), 393-426
MSC (2000): Primary 35C99, 35F05, 35F15, 42B05, 46E35
Published electronically: November 26, 2002
MathSciNet review: 1949165
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Abstract: The main result is the following. Let $\Omega $ be a bounded Lipschitz domain in $\mathbb{R} ^{d}$, $d\geq 2$. Then for every $f\in L^{d}(\Omega )$ with $\int f =0$, there exists a solution $u\in C^{0}(\bar \Omega )\cap W^{1, d}(\Omega )$ of the equation div $u=f$ in $\Omega $, satisfying in addition $u=0$ on $\partial \Omega $ and the estimate

\begin{displaymath}\Vert u\Vert _{L^{\infty }}+\Vert u\Vert _{W^{1, d}}\leq C\Vert f\Vert _{L^{d}} \end{displaymath}

where $C$ depends only on $\Omega $. However one cannot choose $u$ depending linearly on $f$.

Our proof is constructive, but nonlinear--which is quite surprising for such an elementary linear PDE. When $d=2$ there is a simpler proof by duality--hence nonconstructive.

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Additional Information

Jean Bourgain
Affiliation: Institute for Advanced Study, Princeton, New Jersey 08540

Haïm Brezis
Affiliation: Analyse Numérique, Université P. et M. Curie, B.C. 187, 4 Pl. Jussieu, 75252 Paris Cedex 05, France
Address at time of publication: Department of Mathematics, Rutgers University, Hill Center, Busch Campus, 110 Frelinghuysen Rd., Piscataway, New Jersey 08854

Keywords: Divergence equations, gradient equations, critical Sobolev norms
Received by editor(s): January 14, 2002
Received by editor(s) in revised form: October 2, 2002
Published electronically: November 26, 2002
Additional Notes: The first author was partially supported by NSF Grant DMS-9801013
The second author was partially sponsored by a European Grant ERB FMRX CT98 0201. He is also a member of the Institut Universitaire de France.
The authors thank C. Fefferman, P. Lax, P. Mironescu, L. Nirenberg, T. Rivière, M. Vogelius and D. Ye for useful comments
Article copyright: © Copyright 2002 American Mathematical Society

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