Geodesic
The Cauchy problem in Sobolev spaces for Dirac operators
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2009), pp. 51-64.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper we consider the Cauchy problem as a typical example of ill-posed boundary value problems. We describe the necessary and sufficient solvability conditions for the Cauchy problem for a Dirac operator $A$ in Sobolev spaces in a bounded domain $D\subset\mathbb R^n$ with piecewise smooth boundary. Namely, we reduce the Cauchy problem for the Dirac operator to the problem of the harmonic extension from a smaller domain to a larger one. Moreover, along with the solvability conditions for the problem, using bases with double orthogonality, we construct a Carleman formula for recovering a function $u$ from the Sobolev space $H^s(D)$, $s\in\mathbb N$, by its values on $\Gamma$ and values $Au$ in $D$, where $\Gamma$ is an open connected subset of the boundary $\partial D$. It is worth pointing out that we impose no assumptions about geometric properties of the domain $D$, except for its connectedness.
Keywords: Cauchy problem, Dirac operators
Mots-clés : Carleman formula. 
@article{IVM_2009_7_a4,
     author = {I. V. Shestakov},
     title = {The {Cauchy} problem in {Sobolev} spaces for {Dirac} operators},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {51--64},
     publisher = {mathdoc},
     number = {7},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2009_7_a4/}
}
TY  - JOUR
AU  - I. V. Shestakov
TI  - The Cauchy problem in Sobolev spaces for Dirac operators
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2009
SP  - 51
EP  - 64
IS  - 7
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2009_7_a4/
LA  - ru
ID  - IVM_2009_7_a4
ER  - 
%0 Journal Article
%A I. V. Shestakov
%T The Cauchy problem in Sobolev spaces for Dirac operators
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2009
%P 51-64
%N 7
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2009_7_a4/
%G ru
%F IVM_2009_7_a4
I. V. Shestakov. The Cauchy problem in Sobolev spaces for Dirac operators. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2009), pp. 51-64. http://geodesic.mathdoc.fr/item/IVM_2009_7_a4/

[1] Bogolyubov N. N., Shirkov D. V., Vvedenie v teoriyu kvantovannykh polei, Nauka, M., 1984, 600 pp. | MR

[2] Booß-Bavnbek B., Wojciechowski K. P., Elliptic boundary problems for Dirac operators, Birkhäuser, Berlin, 1993, 308 pp. | MR | Zbl

[3] Solomyak M. Z., “O lineinykh ellipticheskikh sistemakh pervogo poryadka”, DAN SSSR, 150:1 (1963), 48–51 | Zbl

[4] Shlapunov A. A., Tarkhanov N. N., “Bases with double orthogonality in the Cauchy problem for systems with injective symbols”, Proc. London Math. Soc., 71:1 (1995), 1–52 | DOI | MR | Zbl

[5] Aizenberg L. A., Kytmanov A. M., “O vozmozhnosti golomorfnogo prodolzheniya v oblast funktsii, zadannykh na kuske ee granitsy”, Matem. sb., 182:4 (1991), 490–507 | MR | Zbl

[6] Lavrentev M. M., Savelev L. Ya., Lineinye operatory i nekorrektnye zadachi, Nauka, M., 1991, 331 pp. | MR

[7] Palamodov V. P., Lineinye differentsialnye operatory s postoyannymi koeffitsientami, Nauka, M., 1967, 488 pp. | MR

[8] Rempel S., Shultse B.-V., Teoriya indeksa ellipticheskikh kraevykh zadach, Mir, M., 1986, 576 pp. | MR | Zbl

[9] Tarkhanov N. N., Metod parametriksa v teorii differentsialnykh kompleksov, Novosibirsk, 1990, 245 pp. | MR

[10] Sobolev S. L., Vvedenie v teoriyu kubaturnykh formul, Nauka, M., 1974, 808 pp. | MR

[11] Shlapunov A. A., “O zadache Koshi dlya operatora Laplasa”, Sib. matem. zhurn., 33:3 (1992), 205–215 | MR | Zbl