Let D be a bounded domain in ℝn (n ≥ 2) with infinitely smooth boundary ∂D. We give some necessary and sufficient conditions for the Cauchy problem to be solvable in the Lebesgue space L 2(D) in D for an arbitrary differential operator A having an injective principal symbol. Furthermore, using bases with double orthogonality, we construct Carleman’s formula that restores a (vector-)function in L 2(D) from the Cauchy data given on a relatively open connected set Γ ⊂ ∂D and the values Au in D whenever the data belong to L 2(Γ) and L 2(D) respectively.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Hadamard J., The Cauchy Problem for Linear Partial Differential Equations of Hyperbolic Type [Russian translation], Nauka, Moscow (1997).
Tarkhanov N. N., The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin (1995).
Lavrent’ev M. M., “On the Cauchy problem for the Laplace equation,” Izv. Akad. Nauk SSSR Ser. Mat., 20, No. 6, 819–842 (1956).
Maz’ya V. G. and Khavin V. P., “On solutions of the Cauchy problem for the Laplace equation (uniqueness, normality, and approximation),” Trudy Moskov. Mat. Obshch., 307, 61–114 (1974).
Kondrat’ev V. A. and Landis E. M., “Qualitative theory of second-order linear partial differential equations,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1988, 32, pp. 99–215 (Itogi Nauki i Tekhniki).
Nacinovich M., “The Cauchy problem for overdetermined systems,” Ann. Mat. Pura Appl. Ser. IV, 156, 265–321 (1990).
Aĭzenberg L. A., Carleman Formulas in Complex Analysis. First Applications [in Russian], Nauka, Novosibirsk (1990).
Aĭzenberg L. A. and Kytmanov A. M., “On the possibility of holomorphic extension into a domain of functions defined on a connected piece of its boundary,” Math. USSR-Sb., 72, No. 2, 467–483 (1992).
Shlapunov A. A. and Tarkhanov N. N., “Bases with double orthogonality in the Cauchy problem for systems with injective symbols,” Proc. London Math. Soc., 71, No. 3, 1–52 (1995).
Shlapunov A. A. and Tarkhanov N. N., “Mixed problems with a parameter,” Ross. Zh. Mat. Fiziki, 12, No. 1, 97–124 (2005).
Shestakov I. V., “The Cauchy problem in Sobolev spaces for Dirac operators,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 51–64 (2009).
Shestakov I. and Shlapunov A., “Negative Sobolev spaces in the Cauchy problem for the Cauchy-Riemann operator,” Zh. SFU Ser. Fiz.-Mat., 2, No. 1, 17–30 (2009).
Fedchenko D. P. and Shlapunov A. A., “On the Cauchy problem for the multi-dimensional Cauchy-Riemann operator in the Lebesgue space L 2 in a domain,” Sb.: Math., 199, No. 11, 1715–1733 (2008).
Hörmander L., Notions of Convexity, Birkhäuser Verlag, Berlin (1994).
Tarkhanov N. N., The Parametrix Method in the Theory of Differential Complexes, Nauka, Novosibirsk (1990).
Shestakov I. and Shlapunov A., “On the Cauchy problem for operators with injective symbols in Sobolev spaces,” Zh. SFU Ser. Fiz.-Mat., 1, No. 1, 52–62 (2008).
Tarkhanov N. N., Laurent Series for Solutions of Elliptic Systems [in Russian], Nauka, Novosibirsk (1991).
Kerzman N., “Hölder and L p-estimates for solutions of \( \bar \partial \) u = f,” Comm. Pure Appl. Math., 24, No. 3, 301–379 (1971).
Hörmander L., “L 2-estimates and existence theorems for the \( \bar \partial \) operator,” Acta Math., 113, No. 1–2, 89–152 (1965).
Straube E. J., “Harmonic and analytic functions admitting a distribution boundary value,” Ann. Sc. Norm. Super. Pisa Cl. Sci., 11, No. 4, 559–591 (1984).
Spencer D. C., “Overdetermined systems of linear partial differential equations,” Bull. Amer. Math. Soc., 75, No. 2, 179–239 (1969).
Dudnikov P. I. and Samborskiĭ S. N., “Overdetermined linear systems of partial differential equations and the related boundary- and initial-value problems,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1991, 65, pp. 5–93 (Itogi Nauki i Tekhniki).
Rempel S. and Schulze B.-W., Index Theory of Elliptic Boundary Problems, Akademie Verlag, Berlin (1982).
Wells R. O., Differential Analysis on Complex Manifolds, Springer-Verlag, New York; Heidelberg; Berlin (1980).
Shlapunov A. A. and Tarkhanov N. N., “On the Cauchy problem for holomorphic functions of Lebesgue class L 2 in domains,” Siberian Math. J., 33, No. 5, 914–922 (1992).
Lavrent’ev M. M., Romanov V. G., and Shishatskiĭ S. P., Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980).
Shapiro H. S., “Stefan Bergman’s theory of doubly-orthogonal functions. An operator-theoretic approach,” Proc. Roy. Irish Acad. Sect. A, 79, No. 6, 49–56 (1979).
Original Russian Text Copyright © 2009 Shestakov I. V. and Shlapunov A. A.
The first author was supported by the Krasnoyarsk Regional Science Foundation (Grant 17G-102) and the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2427.2008.1). The second author was supported by the Siberian Federal University and the Russian Foundation for Basic Research (Grant 08-01-00844).
Krasnoyarsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 687–702, May–June, 2009.
About this article
Cite this article
Shestakov, I.V., Shlapunov, A.A. The Cauchy problem for operators with injective symbol in the Lebesgue space L 2 in a domain. Sib Math J 50, 547–559 (2009). https://doi.org/10.1007/s11202-009-0061-0
- ill-posed Cauchy problem
- Carleman’s formulas