On the Cauchy problem for the elliptic complexes in spaces of distributions


Тип публикации: статья из журнала

Год издания: 2014

Идентификатор DOI: 10.1080/17476933.2013.776043

Ключевые слова: elliptic differential complexes, ill-posed Cauchy problem, Carleman's formula, Primary 35N05, 58J10, Secondary 35J56, 46F12

Аннотация: Let D be a bounded domain in R-n, n >= 2, with a smooth boundary partial derivative D. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A(i)} of first order operators. In particular, we describe traces on partial derivative D of the tangeПоказать полностьюntial part tau(i) (u) and the normal part v(i) (u) of a (vector)-function u from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue space L-2(D), we construct the approximate and exact solutions to the Cauchy problem with the maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman's formulae for a (vector-) function u from the Sobolev space H-1(D) by its Cauchy data tau(i)(u) on a subset Gamma subset of partial derivative D and the values of A(i)u in D modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.

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Выпуск журнала: Vol. 59, Is. 5

Номера страниц: 651-679

ISSN журнала: 17476933

Место издания: ABINGDON



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