On the Cauchy problem for operators with injective symbols in the spaces of distributions

Описание

Тип публикации: статья из журнала

Год издания: 2011

Идентификатор DOI: 10.1515/JIIP.2011.026

Ключевые слова: Ill-posed Cauchy problem, elliptic operators, Carleman's formula

Аннотация: Let D be a bounded domain in the n-dimensional Euclidian space (n >= 2) having smooth boundary partial derivative D. We indicate appropriate Sobolev spaces with negative smoothness in D in order to consider the non-homogeneous ill-posed Cauchy problem for an overdetermined operator A with injective symbol. We prove that elements ofПоказать полностьюthe indicated Sobolev spaces have traces on the boundary. This easily leads to a weak formulation of the Cauchy problem and to the corresponding uniqueness theorem. We also describe solvability conditions of the problem and construct its exact and approximate solutions. Namely, we obtain the Carleman formula recovering a vector-function u from the indicated negative Sobolev class via its Cauchy data on an open connected set Gamma subset of partial derivative D and values of Au on the domain D. Some instructive examples are considered.

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Издание

Журнал: JOURNAL OF INVERSE AND ILL-POSED PROBLEMS

Выпуск журнала: Vol. 19, Is. 1

Номера страниц: 127-150

ISSN журнала: 09280219

Место издания: BERLIN

Издатель: WALTER DE GRUYTER & CO

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