"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."

本文研究一类带变系数的N×N阶椭圆方程组的最优控制问题，该方程组满足共轭条件与涉及薛定谔算子（Schrödinger operator）的混合边界条件。本文的首要研究目标是为该方程组建立极值原理。通过运用格林公式（Green's formula）、第一特征值的相关性质以及Lax-Milgram引理（Lax-Milgram lemma），本文证明了解的存在性与唯一性。此外，本文还严格分析了所考虑的带变系数线性椭圆方程组的极值原理成立的充要条件，以及其解的存在性条件。