Necessary conditions for minimax control problems of second order elliptic partial differential equations (Q1321780)

The paper discusses the problem of the minimizing of \(\text{ess}_{x\in \Omega} \sup h(x,y(u; x), u(x))\) over all pairs \((y,u)\) satisfying \(\Delta u= f(x, y(x), u(x))\), \(x\in \Omega\), \(y\mid_{\partial \Omega} =0\), with additional constraints on the \(y\) (for instance, \(y(x_ i) =a_ i\), \(i=1,\dots, m)\). Under some regularity and growth conditions on the functions \(h\) and \(f\) and some general assumptions on the structure of the problem the author proves an analogue of the Pontryagin's maximum principle.