no. 2
When is a pseudo-differential equation solvable ?
Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 443-460.

Nous commençons par décrire l’état des connaissances sur les problèmes de résolubilité pour les équations aux dérivées partielles et les équations pseudo-différentielles. Nous prouvons ensuite un lemme hilbertien, que nous utilisons pour démontrer un résultat nouveau de résolubilité.

This paper begins with a broad survey of the state of the art in matters of solvability for differential and pseudo-differential equations. Then we proceed with a Hilbertian lemma which we use to prove a new solvability result.

@article{AIF_2000__50_2_443_0,
     author = {Lerner, Nicolas},
     title = {When is a pseudo-differential equation solvable ?},
     journal = {Annales de l'Institut Fourier},
     pages = {443--460},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {50},
     number = {2},
     year = {2000},
     doi = {10.5802/aif.1761},
     mrnumber = {2001e:35188},
     zbl = {0952.35166},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.1761/}
}
TY  - JOUR
AU  - Lerner, Nicolas
TI  - When is a pseudo-differential equation solvable ?
JO  - Annales de l'Institut Fourier
PY  - 2000
SP  - 443
EP  - 460
VL  - 50
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.1761/
DO  - 10.5802/aif.1761
LA  - en
ID  - AIF_2000__50_2_443_0
ER  - 
%0 Journal Article
%A Lerner, Nicolas
%T When is a pseudo-differential equation solvable ?
%J Annales de l'Institut Fourier
%D 2000
%P 443-460
%V 50
%N 2
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.1761/
%R 10.5802/aif.1761
%G en
%F AIF_2000__50_2_443_0
Lerner, Nicolas. When is a pseudo-differential equation solvable ?. Annales de l'Institut Fourier, Tome 50 (2000) no. 2, pp. 443-460. doi : 10.5802/aif.1761. http://archive.numdam.org/articles/10.5802/aif.1761/

[BF] R. Beals, C. Fefferman, On local solvability of linear partial differential equations, Ann. of Math., 97 (1973), 482-498. | MR | Zbl

[Bo] J.M. Bony, Caractérisation des opérateurs pseudo-différentiels, Séminaire EDP, École polytechnique, exposé 23, 1996-1997. | Numdam | Zbl

[BC] J.M. Bony, J.Y. Chemin, Espaces fonctionnels associés au calcul de Weyl-Hörmander, Bull. SMF, 122 (1994), 77-118. | Numdam | MR | Zbl

[D1] N. Dencker, The solvability of nonsolvable operators, Saint-Jean-de-Monts meeting, 1996. | Numdam | MR | Zbl

[D2] N. Dencker, A class of solvable operators, "Geometrical optics and related topics", PNLDE series, volume 32, Birkhäuser, editors F. Colombini, N. Lerner, 1997. | Zbl

[D3] N. Dencker, Estimates and solvability, Arkiv för Matematik, 37, n° 2 (1999), 221-243. | Zbl

[H1] L. Hörmander, Differential equations without solutions, Math. Ann., 140 (1960), 169-173. | MR | Zbl

[H2] L. Hörmander, The analysis of linear partial differential operators, Springer-Verlag, 1985. | Zbl

[H3] L. Hörmander, On the solvability of pseudodifferential equations, pp. 183-213, "Structure of solutions of differential equations", World Scientific Singapore, New Jersey, London, Hong-Kong, editors M. Morimoto and T. Kawai, 1996. | MR | Zbl

[L1] N. Lerner, Sufficiency of condition (ψ) for local solvability in two dimensions, Ann. of Math., 128 (1988), 243-258. | MR | Zbl

[L2] N. Lerner, An iff solvability condition for the oblique derivative problem, Séminaire EDP, École Polytechnique, exposé 18, 1990-1991. | EuDML | Numdam | MR | Zbl

[L3] N. Lerner, Nonsolvability in L2 for a first order operator satisfying condition (ψ), Ann. of Math., 139 (1994), 363-393. | MR | Zbl

[L4] N. Lerner, The Wick calculus of pseudo-differential operators and energy estimates, "New trends in microlocal analysis", editors J.-M. Bony, M. Morimoto, Springer-Verlag, 1997. | MR | Zbl

[L5] N. Lerner, Energy methods via coherent states and advanced pseudo-differential calculus, "Multidimensional complex analysis and partial differential equations", editors P.D. Cordaro, H. Jacobowitz, S. Gindikin, AMS, 1997. | MR | Zbl

[L6] N. Lerner, Perturbation and energy estimates, Ann. Sc. ENS, 31 (1998), 843-886. | EuDML | Numdam | MR | Zbl

[Lw] H. Lewy, An example of a smooth linear partial differential equation without solution, Ann. of Math., 66, 1 (1957) 155-158. | MR | Zbl

[Mi] S. Mizohata, Solutions nulles et solutions non analytiques, J. Math. Kyoto Univ., 1 (1962), 272-302. | MR | Zbl

[NT1] L. Nirenberg, F. Treves, Solvability of a first order linear partial differential equation, Comm. Pure Appl. Math., 16 (1963), 331-351. | MR | Zbl

[NT2] L. Nirenberg, F. Treves, On local solvability of linear partial differential equations, Comm. Pure Appl. Math., 23 (1970), 1-38 and 459-509; 24 (1971), 279-288. | MR | Zbl

[S] H. Smith, An elementary proof of local solvability in two dimensions under condition (ψ), Ann. of Math., 136 (1992), 335-337. | MR | Zbl

Cité par Sources :