Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi

We prove a conjecture by De Giorgi, which states that global weak solutions of nonlinear wave equations such as can be obtained as limits of functions \that minimize suitable functionals of the calculus of variations. These functionals, which are integrals in space-time of a convex Lagrangian, contain an exponential weight with a parameter , and the initial data of the wave equation serve as boundary conditions. As tends to zero, the minimizers converge, up to subsequences, to a solution of the nonlinear wave equation. There is no restriction on the nonlinearity exponent, and the method is easily extended to more general equations.