The De Giorgi approach to Hyperbolic Cauchy problems: an extension to Nonhomogeneous equations

In the present dissertation we discuss an extension of some results obtained by E. Serra and P. Tilli concerning an original conjecture by E. De Giorgi on a purely minimization approach to the Cauchy problem for the defocusing nonlinear wave equation. Precisely, we show how to adapt the techniques developed by Serra and Tilli for homogeneous hyperbolic nonlinear PDEs to the nonhomogeneous case, thus proving that the idea of De Giorgi is in fact an effective approach to investigate general hyperbolic equations. This approach consists of minimizing a functional defined ad hoc, which depends on a small positive parameter ε, and then proving that the sequence of minimizers thus generated admits, up to subsequences, a limit function which satisfies in the sense of distributions the aimed hyperbolic Cauchy problem.