Minima in elliptic variational problems without convexity assumptions (Q1966215)

The paper considers an abstract optimal control problem of the type \[ I(L_1y)+ H(u)\to \min, \] \[ Ay= f+ Bu,\quad I_1(L_2y)\leq 0,\quad F(u)\leq 0,\tag{1} \] \[ u\in L_2(\Sigma; R^k),\quad y\in V, \] where \(\Sigma\) is a bounded set, \(V\) is a Hilbert space, \(A: V\to V'\) is a linear coercive operator, \(B\) is a linear operator, \(L_1\) and \(L_2\) are compact operators, the functionals \(I\), \(I_1\), \(F\) are concave (\(F\) being weakly lower-semicontinuous), but \[ H(u)= \int_\Sigma h(x,u(x)) d\mu \] with a nonconvex integrand \(h\). The author shows how to use the concavity of the functionals \(I\), \(I_1\), \(F\) and the Lyapunov's theorem to obtain the existence of a solution of the problem (1). General results are illustrated on examples with a second-order elliptic operator \(A\).