On the solvability of the equation div u = f in L 1 and in C 0

Bernard Dacorogna; Nicola Fusco; Luc Tartar

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2003)

  • Volume: 14, Issue: 3, page 239-245
  • ISSN: 1120-6330

Abstract

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We show that the equation div u = f has, in general, no Lipschitz (respectively W 1 , 1 ) solution if f is C 0 (respectively L 1 ).

How to cite

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Dacorogna, Bernard, Fusco, Nicola, and Tartar, Luc. "On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 14.3 (2003): 239-245. <http://eudml.org/doc/252292>.

@article{Dacorogna2003,
abstract = {We show that the equation div $u = f$ has, in general, no Lipschitz (respectively $W^\{1,1\}$) solution if $f$ is $C^\{0\}$ (respectively $L^\{1\}$).},
author = {Dacorogna, Bernard, Fusco, Nicola, Tartar, Luc},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Divergence; Lorentz spaces; Sobolev imbedding theorem; divergence},
language = {eng},
month = {9},
number = {3},
pages = {239-245},
publisher = {Accademia Nazionale dei Lincei},
title = {On the solvability of the equation div $u = f$ in $L^\{1\}$ and in $C^\{0\}$},
url = {http://eudml.org/doc/252292},
volume = {14},
year = {2003},
}

TY - JOUR
AU - Dacorogna, Bernard
AU - Fusco, Nicola
AU - Tartar, Luc
TI - On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2003/9//
PB - Accademia Nazionale dei Lincei
VL - 14
IS - 3
SP - 239
EP - 245
AB - We show that the equation div $u = f$ has, in general, no Lipschitz (respectively $W^{1,1}$) solution if $f$ is $C^{0}$ (respectively $L^{1}$).
LA - eng
KW - Divergence; Lorentz spaces; Sobolev imbedding theorem; divergence
UR - http://eudml.org/doc/252292
ER -

References

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  1. BERGH, J. - LÖFSTRÖ, J., Interpolation spaces: an introduction. Springer-Verlag, Berlin1976. Zbl0344.46071MR482275
  2. BOURGAIN, J. - BREZIS, H., Sur l’équation div u = f . C. R. Acad. Sci. Paris, ser. I, 334, 2002, 973-976. Zbl0999.35020MR1913720
  3. GILBARG, D. - TRUDINGER, N.S., Elliptic partial differential equations of second order. Springer-Verlag, Berlin1977. Zbl0562.35001MR473443
  4. MC MULLEN, C.T., Lipschitz maps and nets in Euclidean space. Geom. Funct. Anal., 8, 1998, 304-314. Zbl0941.37030MR1616159DOI10.1007/s000390050058
  5. ORNSTEIN, D., A non-inequality for differential operators in the L 1 norm. Arch. Ration. Mech. Anal., 11, 1962, 40-49. Zbl0106.29602MR149331
  6. PREISS, D., Additional regularity for Lipschitz solutions of PDE. J. Reine Angew. Math., 485, 1997, 197-207. Zbl0870.35022MR1442194DOI10.1515/crll.1997.485.197
  7. STEIN, E.M., Singular integrals and differentiability properties of functions. Princeton University Press, Princeton1970. Zbl0207.13501MR290095
  8. STEIN, E.M. - WEISS, G., Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton1971. Zbl1026.42001MR304972
  9. TARTAR, L., Imbedding theorems of Sobolev spaces into Lorentz spaces. Bollettino UMI, 1-B, 1998, 479-500. Zbl0929.46028MR1662313

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