Spectral Properties and Boundary Conditions for Second Order Elliptic Operators

Abstract

This article investigates the regular solvability of boundary value problems for second-order elliptic operator-differential equations with discontinuous coefficients and non-standard boundary conditions in a Hilbert space setting. The authors focus on operator equations defined over a finite interval and aim to identify the structural and spectral conditions under which these problems admit unique, stable solutions. They construct the problem using self-adjoint and positively defined operators, incorporating piecewise-constant coefficients and boundary operators. The study introduces a specialized function space to accommodate the differential and boundary conditions and establishes conditions ensuring the existence, uniqueness, and continuity of solutions with respect to input data. By analyzing the properties of associated linear operators and leveraging classical results from functional analysis, such as the Banach inverse operator theorem, the authors demonstrate that the solution operator is bounded and invertible under specific spectral constraints. This work contributes to the theoretical understanding of elliptic operator-differential equations and provides valuable tools for further analysis in mathematical physics, engineering, and applied mathematics contexts where such boundary value problems frequently arise.