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.
@article{10_4007_annals_2012_175_3_11,
author = {Enrico Serra and Paolo Tilli},
title = {Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by {De} {Giorgi}},
journal = {Annals of mathematics},
pages = {1551--1574},
year = {2012},
volume = {175},
number = {3},
doi = {10.4007/annals.2012.175.3.11},
mrnumber = {2912711},
zbl = {06051276},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.3.11/}
}
TY - JOUR AU - Enrico Serra AU - Paolo Tilli TI - Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi JO - Annals of mathematics PY - 2012 SP - 1551 EP - 1574 VL - 175 IS - 3 UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.3.11/ DO - 10.4007/annals.2012.175.3.11 LA - en ID - 10_4007_annals_2012_175_3_11 ER -
%0 Journal Article %A Enrico Serra %A Paolo Tilli %T Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi %J Annals of mathematics %D 2012 %P 1551-1574 %V 175 %N 3 %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2012.175.3.11/ %R 10.4007/annals.2012.175.3.11 %G en %F 10_4007_annals_2012_175_3_11
Enrico Serra; Paolo Tilli. Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi. Annals of mathematics, Tome 175 (2012) no. 3, pp. 1551-1574. doi: 10.4007/annals.2012.175.3.11
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