  In this talk we present an overview on the extensions of the De Giorgi
approach to general second order nonlinear hyperbolic equations. We start with
an introduction to the original conjecture by E. De Giorgi (De Giorgi '96) and
to its solution by E. Serra and P. Tilli (Serra&Tilli '12). Then, we discuss a
first extension of this idea (Serra&Tilli '16) aimed at investigating a wide
class of homogeneous equations. Finally, we announce a further extension to
nonhomogeneous equations, obtained by the author in collaboration with P. Tilli
(Tentarelli&Tilli '17).
