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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the equation $\operatorname {div}Y=f$ and application to control of phases
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by Jean Bourgain and Haïm Brezis PDF
J. Amer. Math. Soc. 16 (2003), 393-426 Request permission

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{equation*}\Vert u\Vert _{L^{\infty }}+\Vert u\Vert _{W^{1, d}}\leq C\Vert f\Vert _{L^{d}} \end{equation*} 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
  • MR Author ID: 40280
  • Email: bourgain@math.ias.edu
  • 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
  • MR Author ID: 41485
  • Email: brezis@ccr.jussieu.fr, brezis@math.rutgers.edu
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 16 (2003), 393-426
  • MSC (2000): Primary 35C99, 35F05, 35F15, 42B05, 46E35
  • DOI: https://doi.org/10.1090/S0894-0347-02-00411-3
  • MathSciNet review: 1949165