This paper deals with second-order optimality conditions and regularity of Lagrange multipliers for a class of optimal control problems governed by semilinear elliptic equations with mixed pointwise constraints in which controls act both in the domain and on the boundary. We give some criteria under which the optimality conditions are of KKT type and the multipliers are of \(L^p\)-spaces. Moreover, we show that the multipliers are Lipschitz continuous functions.

中文翻译：

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混合最优控制问题的二阶最优条件和拉格朗日乘子的正则性

本文讨论了由半线性椭圆方程控制的一类最优控制问题的拉格朗日乘子的二阶最优性条件和正则性，其中控制既在域中又在边界上起作用。我们给出了一些标准，在这些标准下，最优条件是 KKT 类型，乘数是\(L^p\) -空间。此外，我们表明乘数是 Lipschitz 连续函数。