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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[DG]
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AUTHOR = {De Giorgi, Ennio},
TITLE = {Conjectures concerning some evolution problems. A celebration of {J}ohn {F}. {N}ash, {J}r.},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {81},
YEAR = {1996},
NUMBER = {2},
PAGES = {255--268},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35K55 (35L70)},
MRNUMBER = {1395405},
MRREVIEWER = {Marco Degiovanni},
DOI = {10.1215/S0012-7094-96-08114-4},
ZBLNUMBER = {0874.35027},
} -
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TITLE = {Selected Papers},
PUBLISHER = {Springer-Verlag},
NOTE={(edited by L. Ambrosio, G. Dal Maso, M. Forti, M. Miranda, and S. Spagnolo)},
ADDRESS = {New York},
YEAR = {2006},
ISBN = {3-540-26169-9; 978-3-540-26169-8},
MRCLASS = {49-03 (00B60 01A75 35-03)},
MRNUMBER = {2229237},
ZBLNUMBER = {1151.68001},
} -
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@incollection {N, MRKEY = {2229237},
AUTHOR = {Nirenberg, L},
TITLE={Remarks on some of the analytic works of {E}nnio {D}e {G}iorgi},
BOOKTITLE = {Selected Papers},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {2006},
PAGES = {38--42},
ISBN = {3-540-26169-9; 978-3-540-26169-8},
MRCLASS = {49-03 (00B60 01A75 35-03)},
MRNUMBER = {2229237},
ZBLNUMBER = {1151.68001},
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AUTHOR = {Stefanelli, Ulisse},
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FJOURNAL = {Mathematical Models and Methods in Applied Sciences},
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PAGES = {1377--1394},
ISSN = {0218-2025},
MRCLASS = {49J45 (35A15 35B25 35L71)},
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ZBLNUMBER = {1106.35001},
}
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