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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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Author information

Correspondence to A. A. Shlapunov.

Additional information

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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Keywords

  • ill-posed Cauchy problem
  • Carleman’s formulas