On the Cauchy Problem for Operators with Injective Symbols in Sobolev Spaces

Ivan V. Shestakov ; Alexander A. Shlapunov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 1 (2008) no. 1, pp. 52-62.

Résumé

Let $D$ be a bounded domain in $\mathbb R^n$ ($n\ge 2$) with a smooth boundary $\partial D$. We describe necessary and sufficient solvability conditions (in Sobolev spaces in $D$) of the ill-posed non-homogeneous Cauchy problem for a partial differential operator $A$ with injective symbol and of order $m\ge 1$. Moreover, using bases with the double orthogonality property we construct Carleman's formulae for (vector-) functions from the Sobolev space $H^s(D)$, $s\ge m$, by their Cauchy data on $\Gamma$ and the values of $Au$ in $D$ where $\Gamma$ is an open (in the topology of $\partial D$) connected part of the boundary.

Détail

Keywords: ill-posed Cauchy problem, bases with double orthogonality.
Mots-clés : Carleman's formula

Bibliographie

[1] J. Hadamard, Le problème de Cauchy et les equations aux derivées partielles linéares hyperboliques, Gauthier-Villars, Paris, 1932 | Zbl

[2] N. N. Tarkhanov, The Cauchy problem for solutions of elliptic equations, Akademie Verlag, Berlin, 1995 | MR | Zbl

[3] M. M. Lavrent'ev, “On the Cauchy problem for Laplace's equation”, Izvestija AN SSSR. Ser. mat., 20 (1956), 819–842 (Russian) | MR

[4] V. G. Maz'ya, V. P. Havin, “On the solutions of the Cauchy problem for Laplace's equation (uniqueness, normality, approximation)”, Trans. Mosc. Math. Soc., 307 (1974), 61–114 (Russian) | MR

[5] V. A. Kondrat'ev, E. M. Landis, “Qualitative theory for linear differential equations of the second order”, Results of Sciences and Technics, Modern Problems of Mathematics, Fundamental Directions, 32, VINITI AN SSSR, Moscow, 1988, 99–215 (Russian) | MR

[6] Kluwer Ac. Publ., 1993 | MR

[7] M. Nacinovich, “Cauchy problem for overdetermined systems”, Ann. di Mat. Pura ed Appl. (IV), 156 (1990), 265–321 | DOI | MR | Zbl

[8] L. A. Aizenberg, A. M. Kytmanov, “On possibility of holomorphic extension to a domain of fuctions, given on a part of its boundary”, Mat. Sb., 182:4 (1991), 490–507 (Russian) | MR | Zbl

[9] A. A. Shlapunov, N. N. Tarkhanov, “Bases with double orthogonality in the Cauchy problem for systems with injective symbols”, Proc. London Math. Soc., 71:3 (1995), 1–54 | DOI | MR

[10] A. A. Shlapunov, N. N. Tarkhanov, “Mixed problems with a parameter”, Russ. J. Math. Phys., 12:1 (2005), 97–124 | MR

[11] L. Hörmander, Notions of convexity, Birkhäuser Verlag, Berlin, 1994 | MR | Zbl

[12] N. N. Tarkhanov, Complexes of differential operators, Kluwer Ac. Publ., Dordrecht, 1995 | MR | Zbl

[13] Springer Verlag, Berlin, 1992 | MR

[14] P. I. Dudnikov, S. N. Samborskii, “Boundary value and initial-boundary value problem for linear overdetermined systems of partial differential equations”, Results of Sciences and Technics, Modern Problems of Mathematics. Fundamental Directions, 65, VINITI, Moscow, 1991, 5–93 (Russian) | MR

[15] N. N. Tarkhanov, Analysis of solutions to elliptic equations, Kluwer Ac. Publ., Dordrecht, 1997 | MR | Zbl

[16] S. Rempel, B.-W. Schulze, Index theory of elliptic boundary problems, Akademie Verlag, Berlin, 1986 | MR

[17] R. Wells, Differential analysis on complex manifolds, Prentice Hall, Englewood Cliffs, N.J., 1973 | MR | Zbl

[18] H. S. Shapiro, “Stefan Bergman's theory of doubly-orthogonal functions. An operator-theoretic approach”, Proc. Roy. Ac. Sect., 79:6 (1979), 49–56 | MR