Abstract:
This paper is devoted to the solvability of boundary value problems in a cylindrical domain and certain spectral problems for a second-order linear elliptic equation with involutive deviation of the argument in lower-order terms with respect to a selected variable. The paper consists of two parts. In the first part we investigate the solvability of boundary value problems, including nonlocal ones, for a second-order linear elliptic equation with variable coefficients and general involutive deviation of the argument with respect to a selected variable. We establish existence and uniqueness theorems for regular solutions (those possessing all generalised derivatives in the Sobolev sense that appear in the equation). In the second part we investigate the solvability of certain spectral problems for an elliptic equation with constant coefficients and linear involutive deviation of the argument with respect to a selected variable. Specifically, we analyse how various parameters affect the uniqueness and non-uniqueness of regular solutions to boundary value problems. These results show that the presence of involution (involutive deviation of the argument) in the equation can substantially impact both the solvability conditions and the well-posedness of the problems.
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