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.

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Authors

Enrico Serra

Dipartimento di Matematica
Politecnico di Torino
Corso Duca degli Abruzzi, 24
10129 Torino
Italy

Paolo Tilli

Dipartimento di Matematica
Politecnico di Torino
Corso Duca degli Abruzzi, 24
10129 Torino
Italy