Optimization problems for nonlinear eigenvalues

This thesis is mainly focused on the study of variational problems and the related elliptic partial differential equations, in which the role usually played by the Euclidean norm is taken by a generic Finslerian norm, whose unit ball is a generic centrally symmetric convex body, called Wulff shape. This kind of problems are called anisotropic problem. We study geometric properties of the eigenvalues of the anisotropic p-Laplacian with Dirichlet or Neumann boundary conditions, where F is a suitable norm. In particular, we find sharp upper and lower bounds for these eigenvalues with respect to an open set. Finally we treat problems associated to non-standard Euler-Lagrange equations, that are called "nonlocal" problems. In particular we study problems where the integral term of the unknown function calculated on the entire domain represents the non-locality.