Nonlinear wave equations as limits of convex minimization problems: proof of a conjecture by De Giorgi
成果类型：
Article
署名作者：
Serra, Enrico; Tilli, Paolo
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2012.175.3.11
发表日期：
2012
页码：
1551-1574
关键词：
摘要：
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 epsilon, and the initial data of the wave equation serve as boundary conditions. As epsilon tends to zero, the minimizers v(epsilon) 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.