In this paper, we investigate the Fredholm solvability of the generalized Neumann problem for a high-order elliptic equation in the plane. The equivalence of the solvability conditions of the generalized Neumann problem to the complementarity condition (Shapiro–Lopatinsky condition) is proved. The formula for the index of the specified problem in the class $C^{2l,\mu }\left(\overline{D}\right)$ is calculated.

中文翻译：

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关于 Fredholm 可解性和椭圆方程的广义 Neumann 问题的指数

在本文中，我们研究了平面上高阶椭圆方程的广义 Neumann 问题的 Fredholm 可解性。证明了广义诺依曼问题的可解性条件与互补性条件（Shapiro-Lopatinsky 条件）的等价性。类中指定问题的索引公式$C^{2l,\mu }\left(\overline{D}\right)$是计算出来的。