On the extensions of the De Giorgi approach to nonlinear hyperbolic equations

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 ([1, 2]) and to its solution by E. Serra and P. Tilli ([4]). Then, we discuss a first extension of this idea (Serra&Tilli, [5]) aimed at investigating a wide class of homogeneous equations. Finally, we announce a further extension to nonhomogeneous equations, obtained by the author in [9] in collaboration with P. Tilli.