On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$

Dacorogna, Bernard; Fusco, Nicola; Tartar, Luc

Geodesic

Parcourir par

On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$

Dacorogna, Bernard ; Fusco, Nicola ; Tartar, Luc

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 3, pp. 239-245.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

Résumé

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

MR Zbl

@article{RLIN_2003_9_14_3_a6,
     author = {Dacorogna, Bernard and Fusco, Nicola and Tartar, Luc},
     title = {On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$},
     journal = {Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni},
     pages = {239--245},
     publisher = {mathdoc},
     volume = {Ser. 9, 14},
     number = {3},
     year = {2003},
     zbl = {1225.35050},
     mrnumber = {482275},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_3_a6/}
}

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. Rendiconti Lincei. Matematica e applicazioni
PY  - 2003
SP  - 239
EP  - 245
VL  - 14
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_3_a6/
LA  - en
ID  - RLIN_2003_9_14_3_a6
ER  -

%0 Journal Article
%A Dacorogna, Bernard
%A Fusco, Nicola
%A Tartar, Luc
%T On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$
%J Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni
%D 2003
%P 239-245
%V 14
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_3_a6/
%G en
%F RLIN_2003_9_14_3_a6

Dacorogna, Bernard; Fusco, Nicola; Tartar, Luc. On the solvability of the equation div $u = f$ in $L^{1}$ and in $C^{0}$. Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 14 (2003) no. 3, pp. 239-245. http://geodesic.mathdoc.fr/item/RLIN_2003_9_14_3_a6/

Bibliographie
Cité par

[1] J. Bergh - J. Löfströ, Interpolation spaces: an introduction. Springer-Verlag, Berlin 1976. | MR | Zbl

[2] J. Bourgain - H. Brezis, Sur l’équation div $u = f$. C. R. Acad. Sci. Paris, ser. I, 334, 2002, 973-976. | MR | Zbl

[3] D. Gilbarg - N.S. Trudinger, Elliptic partial differential equations of second order. Springer-Verlag, Berlin 1977. | MR | Zbl

[4] C.T. Mc Mullen, Lipschitz maps and nets in Euclidean space. Geom. Funct. Anal., 8, 1998, 304-314. | DOI | MR | Zbl

[5] D. Ornstein, A non-inequality for differential operators in the $L^{1}$ norm. Arch. Ration. Mech. Anal., 11, 1962, 40-49. | MR | Zbl

[6] D. Preiss, Additional regularity for Lipschitz solutions of PDE. J. Reine Angew. Math., 485, 1997, 197-207. | fulltext EuDML | DOI | MR | Zbl

[7] E.M. Stein, Singular integrals and differentiability properties of functions. Princeton University Press, Princeton 1970. | MR | Zbl

[8] E.M. Stein - G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton 1971. | MR | Zbl