On the Existence of Weak Optimal Controls in the Coefficients for a Degenerate Anisotropic p-Laplacian

Abstract

We consider an optimal control problem for nonlinear degenerate elliptic problems involving an anisotropic p-Laplacian and Dirichlet boundary conditions. We take the matrix-valued coefficients A(x) of such system as a control in \(L^{p/2}(\varOmega ;\mathbb {R}^{\frac{N(N+1)}{2}})\). One of the important features of the admissible controls is the fact that eigenvalues of the coefficient matrices may vanish in \(\varOmega \). Equations of this type may exhibit the Lavrentiev phenomenon and nonuniqueness of weak solutions. Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of this optimal control problem in the class of weak solutions.