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 w + |w|p−2w = 0 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 ε, and the initial data of the wave equation serve as boundary conditions. As ε tends to zero, the minimizers vε 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.