Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solu- tions are obtained as limits of functions that minimize suitable functionals in space{time (where the initial data of the Cauchy Problem serve as pre- scribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the Calculus of Variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that t in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.

A minimization approach to hyperbolic Cauchy problems / Serra, Enrico; Tilli, Paolo. - In: JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY. - ISSN 1435-9855. - STAMPA. - 18:9(2016), pp. 2019-2044. [10.4171/JEMS/637]

A minimization approach to hyperbolic Cauchy problems

SERRA, Enrico;TILLI, PAOLO
2016

Abstract

Developing an original idea of De Giorgi, we introduce a new and purely variational approach to the Cauchy Problem for a wide class of defocusing hyperbolic equations. The main novel feature is that the solu- tions are obtained as limits of functions that minimize suitable functionals in space{time (where the initial data of the Cauchy Problem serve as pre- scribed boundary conditions). This opens up the way to new connections between the hyperbolic world and that of the Calculus of Variations. Also dissipative equations can be treated. Finally, we discuss several examples of equations that t in this framework, including nonlocal equations, in particular equations with the fractional Laplacian.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2651082