Variational characterizations of weak solutions to the Dirichlet problem for mixed-type equations

For linear and nonlinear second-order partial differential equations of mixed elliptic-hyperbolic type, we prove that weak solutions to the Dirichlet problem are characterized by a variational principle. The weak solutions are shown to be saddle points of natural functionals suggested by the divergence form of the PDEs. Moreover, the natural domains of the functionals are the weighted Sobolev spaces to which the solutions belong. In addition, all critical levels will be characterized in terms of global extrema of the functional restricted to suitable infinite-dimensional linear subspaces. These subspaces are defined in terms of a robust spectral theory with weights associated to the linear operator. This spectral theory has been recently developed by the authors, which in turn exploits weak well-posedness results obtained by Morawetz and the authors.