Transmission problems for elliptic equations with variable coefficients and Schr\u00f6dinger operators on the interface

This paper investigates the optimal control of an N\u00d7N elliptic system with variable coefficients, subject to conjugation conditions and mixed boundary conditions involving a Schr\u00f6dinger operator. The primary objective is to establish maximum principles for this system. By utilizing Green's formula, properties of the first eigenvalue, and the Lax-Milgram lemma, the existence and uniqueness of solutions are demonstrated. Furthermore, the necessary and sufficient conditions for the validity of the maximum principle and the existence of solutions to the considered linear elliptic system with variable coefficients are rigorously analyzed.