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Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi. (English) Zbl 1251.49019
Summary: 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.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35L05 Wave equation
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