A minimization approach to hyperbolic Cauchy problems

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 solutions are obtained as limits of functions that minimize suitable functionals in spacetime (where the initial data of the Cauchy problem serve as prescribed 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 fit into this framework, including nonlocal equations, in particular equations with the fractional Laplacian.