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

Abstract

We prove a conjecture by De Giorgi, which states that global weak solutions of nonlinear wave equations such as $\square w+|w|^{p-2}w=0$ 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 $\varepsilon$, and the initial data of the wave equation serve as boundary conditions. As $\varepsilon$ tends to zero, the minimizers $v_\varepsilon$ 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.