On an Optimal L¹-Control Problem in Coefficients for Linear Elliptic Variational Inequality

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogenous Dirichlet boundary conditions. We take the matrix-valued coefficients A(x) in the main part of the elliptic operator as controls in L1(Ω;R N(N+1)/2). Since the eigenvalues of such matrices may vanish and be unbounded in Ω, it leads to the “non-coercivity trouble”. Using the concept of convergence in variable spaces and following the direct method in the Calculus of Variations, we establish the solvability of the optimal control problem in the class of the so called H-admissible solutions.