A criterion of solvability of the elliptic Cauchy problem in a multi-dimensional cylindrical domain

ABSTRACT

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.

KEYWORDS:

• Elliptic Cauchy problem
• self-adjoint operator
• elliptic operator with deviating argument

AMS SUBJECT CLASSIFICATIONS:

• 31A30
• 31B30
• 35J40

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This publication is financially supported by a grant from the Ministry of Science and Education of the Republic of Kazakhstan [grant number AP05133239].