The paper deals with the three-dimensional Robin type boundary-value problem (BVP) for a second-order strongly elliptic system of partial differential equations in the divergence form with variable coefficients. The problem is studied by the localized parametrix based potential method. By using Green’s representation formula and properties of the localized layer and volume potentials, the BVP under consideration is reduced to the a system of localized boundary-domain singular integral equations (LBDSIE). The equivalence between the original boundary value problem and the corresponding LBDSIE system is established. The matrix operator generated by the LBDSIE system belongs to the Boutet de Monvel algebra. With the help of the Vishik–Eskin theory based on the Wiener–Hopf factorization method, the Fredholm properties of the corresponding localized boundary-domain singular integral operator are investigated and its invertibility in appropriate function spaces is proved.

中文翻译：

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自伴二阶强椭圆PDE系统Robin类型问题的局部边界域奇异积分方程

针对具有变系数散度形式的二阶强椭圆型偏微分方程组，研究了三维Robin型边值问题（BVP）。通过基于局部参数的势能方法研究了该问题。通过使用格林的表示公式以及局部化层的性质和体积电势，考虑中的BVP简化为局部化的边界域奇异积分方程组（LBDSIE）。建立了原始边值问题与相应的LBDSIE系统之间的等价关系。LBDSIE系统生成的矩阵运算符属于Boutet de Monvel代数。借助基于Wiener-Hopf因式分解方法的Vishik-Eskin理论，