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On the regularization Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain. (English) Zbl 1486.35149

Summary: In this paper, the problem of continuation of the solution of the ill-posed Cauchy problem for matrix factorizations of the Helmholtz equation in a multidimensional bounded domain is considered. It is assumed that the solution to the problem exists and is continuously differentiable in a closed domain with exactly given Cauchy data. For this case, an explicit formula for the continuation of the solution is established, as well as a regularization formula for the case where, under the indicated conditions, instead of Cauchy data their continuous approximations with a given error in uniform metric are given. A stability estimate for the solution of the Cauchy problem in the classical sense is obtained.

MSC:

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J56 Boundary value problems for first-order elliptic systems
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