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The Cauchy problem for operators with injective symbol in the Lebesgue space L 2 in a domain

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Abstract

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.

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References

  1. Hadamard J., The Cauchy Problem for Linear Partial Differential Equations of Hyperbolic Type [Russian translation], Nauka, Moscow (1997).

    Google Scholar 

  2. Tarkhanov N. N., The Cauchy Problem for Solutions of Elliptic Equations, Akademie Verlag, Berlin (1995).

    MATH  Google Scholar 

  3. Lavrent’ev M. M., “On the Cauchy problem for the Laplace equation,” Izv. Akad. Nauk SSSR Ser. Mat., 20, No. 6, 819–842 (1956).

    MATH  MathSciNet  Google Scholar 

  4. 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).

    Google Scholar 

  5. 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).

    Google Scholar 

  6. Nacinovich M., “The Cauchy problem for overdetermined systems,” Ann. Mat. Pura Appl. Ser. IV, 156, 265–321 (1990).

    Article  MATH  MathSciNet  Google Scholar 

  7. Aĭzenberg L. A., Carleman Formulas in Complex Analysis. First Applications [in Russian], Nauka, Novosibirsk (1990).

    Google Scholar 

  8. 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).

    Article  Google Scholar 

  9. 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).

    Article  MATH  MathSciNet  Google Scholar 

  10. Shlapunov A. A. and Tarkhanov N. N., “Mixed problems with a parameter,” Ross. Zh. Mat. Fiziki, 12, No. 1, 97–124 (2005).

    MATH  MathSciNet  Google Scholar 

  11. Shestakov I. V., “The Cauchy problem in Sobolev spaces for Dirac operators,” Izv. Vyssh. Uchebn. Zaved. Mat., No. 7, 51–64 (2009).

  12. 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).

    Google Scholar 

  13. 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).

    Article  MATH  MathSciNet  Google Scholar 

  14. Hörmander L., Notions of Convexity, Birkhäuser Verlag, Berlin (1994).

    MATH  Google Scholar 

  15. Tarkhanov N. N., The Parametrix Method in the Theory of Differential Complexes, Nauka, Novosibirsk (1990).

    MATH  Google Scholar 

  16. 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).

    Google Scholar 

  17. Tarkhanov N. N., Laurent Series for Solutions of Elliptic Systems [in Russian], Nauka, Novosibirsk (1991).

    Google Scholar 

  18. Kerzman N., “Hölder and L p-estimates for solutions of \( \bar \partial \) u = f,” Comm. Pure Appl. Math., 24, No. 3, 301–379 (1971).

    Article  MathSciNet  Google Scholar 

  19. Hörmander L., “L 2-estimates and existence theorems for the \( \bar \partial \) operator,” Acta Math., 113, No. 1–2, 89–152 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  20. 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).

    MATH  MathSciNet  Google Scholar 

  21. Spencer D. C., “Overdetermined systems of linear partial differential equations,” Bull. Amer. Math. Soc., 75, No. 2, 179–239 (1969).

    Article  MATH  MathSciNet  Google Scholar 

  22. 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).

    Google Scholar 

  23. Rempel S. and Schulze B.-W., Index Theory of Elliptic Boundary Problems, Akademie Verlag, Berlin (1982).

    MATH  Google Scholar 

  24. Wells R. O., Differential Analysis on Complex Manifolds, Springer-Verlag, New York; Heidelberg; Berlin (1980).

    MATH  Google Scholar 

  25. 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).

    Article  MathSciNet  Google Scholar 

  26. Lavrent’ev M. M., Romanov V. G., and Shishatskiĭ S. P., Ill-Posed Problems of Mathematical Physics and Analysis [in Russian], Nauka, Moscow (1980).

    Google Scholar 

  27. 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).

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Siberian Federal University, Krasnoyarsk, Russia

    I. V. Shestakov & A. A. Shlapunov

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  1. I. V. Shestakov
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  2. A. A. Shlapunov
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Correspondence to A. A. Shlapunov.

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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).

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Krasnoyarsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 687–702, May–June, 2009.

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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

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