Necessary and sufficient conditions for discrete and differential inclusions of elliptic type (Q855397)

The following two optimization problems are considered for discrete elliptic inclusions: \[ \begin{aligned} \text{minimize}\qquad &\sum_{x_1=1,\dots,T-1,x_2=1,\dots,L-1} g_{x_1,x_2}(u_{x_1,x_2}) \\ \text{subject to}\qquad &u_{x_1+1,x_2}\in F_{x_1,x_2} (u_{x_1-1,x_2},u_{x_1,x_2-1}, u_{x_1,x_2},u_{x_1,x_2+1})\\ \text{and}\qquad &u_{x_1,0}=\alpha_{0x_1},\: u_{x_1,L}=\alpha_{Lx_1}, \: u_{0,x_2}=\beta_{0x_2},\: u_{T,x_2}=\beta_{Tx_2},\\ &x_1=1,\ldots,T-1;\quad x_2=1,\ldots,L-1, \end{aligned}\tag{\(P_D\)} \] and for elliptic differential inclusions: \[ \begin{aligned} \text{minimize}\qquad &J(u(\cdot)):=\int\int_R g(u(x),x)\,dx \\ \text{subject to}\qquad &\Delta u(x)\in F(u(x),x),\quad x\in R\\ \text{and}\qquad &u(x)=\beta(x),\quad x\in B, \end{aligned}\tag{\(P_C\)} \] where for the problem \((P_D)\) \(g_{x_1,x_2}:\mathbb{R}^n\to\mathbb{R}\cup\{\pm\infty\},\) \(F_{x_1,x_2}:\mathbb{R}^{4n}\to 2^{\mathbb{R}^n}\) are multivalued mappings, \(\alpha_{0x_1},\) \(\alpha_{Lx_1},\) \(\beta_{0x_2},\) \(\beta_{Tx_2}\) are fixed vectors, \(T\) and \(L\) are natural numbers, and in the case of the problem \((P_C)\) the function \(g:\mathbb{R}^n\times R\to\mathbb{R}\) is given, \(R\subset\mathbb{R}^2\) is bounded region with closed piecewise-smooth simple boundary \(B\), \(\beta\) is a continuous function and \(F(\cdot,x):\mathbb{R}^n\to 2^{\mathbb{R}^n}\) is a multivalued mapping for all fixed \(x\in R.\) Here \(\Delta\) denotes the Laplace operator. A new concept of locally adjoint mappings is introduced. It differs from the definition given in [\textit{B. S. Mordukhovich}, Approximation methods in problems of optimization and control. Nauka, Moscow (1988; Zbl 0643.49001)]. Necessary and sufficient conditions for optimality problems \((P_D)\) and are obtained. In the case when \(n=1\) the results for problem \((P_C)\) are extended for multidimensional regions and general second order elliptic operators instead of the Laplace operator.