Comptes Rendus
Partial differential equations/Functional analysis
On the regularity of solutions to Poisson's equation
[Sur la régularité des solutions de l'équation de Poisson]
Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 819-823.

Dans cette note, nous annonçons de nouveaux résultats de régularité pour des solutions distributionelles localement intégrables à l'équation de Poisson. Cela comprend, par exemple, les solutions standard obtenues par convolution avec la solution fondamentale. En particulier, nos résultats montrent qu'il n'y a aucune différence qualitative de régularité entre ces solutions dans le plan et celles en dimensions supérieures.

In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In particular, our results show that there is no qualitative difference in the regularity of these solutions in the plane and in higher dimensions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.07.001

Rahul Garg 1 ; Daniel Spector 1, 2

1 Technion – Israel Institute of Technology, Department of Mathematics, Haifa, Israel
2 National Chiao Tung University, Department of Applied Mathematics, Hsinchu, Taiwan
@article{CRMATH_2015__353_9_819_0,
     author = {Rahul Garg and Daniel Spector},
     title = {On the regularity of solutions to {Poisson's} equation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {819--823},
     publisher = {Elsevier},
     volume = {353},
     number = {9},
     year = {2015},
     doi = {10.1016/j.crma.2015.07.001},
     language = {en},
}
TY  - JOUR
AU  - Rahul Garg
AU  - Daniel Spector
TI  - On the regularity of solutions to Poisson's equation
JO  - Comptes Rendus. Mathématique
PY  - 2015
SP  - 819
EP  - 823
VL  - 353
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crma.2015.07.001
LA  - en
ID  - CRMATH_2015__353_9_819_0
ER  - 
%0 Journal Article
%A Rahul Garg
%A Daniel Spector
%T On the regularity of solutions to Poisson's equation
%J Comptes Rendus. Mathématique
%D 2015
%P 819-823
%V 353
%N 9
%I Elsevier
%R 10.1016/j.crma.2015.07.001
%G en
%F CRMATH_2015__353_9_819_0
Rahul Garg; Daniel Spector. On the regularity of solutions to Poisson's equation. Comptes Rendus. Mathématique, Volume 353 (2015) no. 9, pp. 819-823. doi : 10.1016/j.crma.2015.07.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.07.001/

[1] J. Bourgain; H. Brezis On the equation divY=f and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003), pp. 393-426

[2] H. Brezis; S. Wainger A note on limiting cases of Sobolev embeddings and convolution inequalities, Commun. Partial Differ. Equ., Volume 5 (1980), pp. 773-789

[3] C. Fefferman; E. Stein Hp spaces of several variables, Acta Math., Volume 129 (1972), pp. 137-193

[4] R. Garg, D. Spector, On the role of Riesz potentials in Poisson's equation and Sobolev embeddings, Indiana U. Math J. (to appear).

[5] B. Jawerth Some observations on Besov and Lizorkin–Triebel spaces, Math. Scand., Volume 40 (1977), pp. 94-104

[6] E.H. Lieb; M. Loss Analysis, American Mathematical Society, Providence, RI, 2001

[7] V. Maz'ya Sobolev Spaces, Springer-Verlag, Berlin, 1985

[8] Y. Mizuta Potential Theory in Euclidean Spaces, Gakkōtosho, Tokyo, 1996

[9] J. Strömberg; R. Wheeden Fractional integrals on weighted Hp and Lp spaces, Trans. Amer. Math. Soc., Volume 287 (1985), pp. 293-321

[10] A. Uchiyama A constructive proof of the Fefferman–Stein decomposition of BMO(Rn), Acta Math., Volume 148 (1982), pp. 215-241

Cité par Sources :

Commentaires - Politique