De Giorgi's approach to hyperbolic Cauchy problems: the case of nonhomogeneous equations

TL;DR

This paper extends De Giorgi's minimization approach to nonhomogeneous hyperbolic Cauchy problems, demonstrating its effectiveness for a broader class of nonlinear wave equations.

Contribution

It generalizes Serra and Tilli's techniques from homogeneous to nonhomogeneous hyperbolic PDEs, validating De Giorgi's approach for more complex equations.

Findings

Extended De Giorgi's method to nonhomogeneous equations

Proved effectiveness of minimization approach for general hyperbolic PDEs

Connected De Giorgi's ideas with recent PDE techniques

Abstract

In this paper we discuss an extension of some results obtained by E. Serra and P. Tilli, in [Serra&Tilli '12, Serra&Tilli '16], concerning an original conjecture by E. De Giorgi ([De Giorgi '96, De Giorgi '06]) on a purely minimization approach to the Cauchy problem for the defocusing nonlinear wave equation. Precisely, we show how to extend the techniques developed by Serra and Tilli for homogeneous hyperbolic nonlinear PDEs to the nonhomogeneous case, thus proving that the idea of De Giorgi yields in fact an effective approach to investigate general hyperbolic equations.