Existence results for a fourth order equation arising in the theory of non-equilibrium phase transitions

In this talk we will introduce a model that arises in the
theory of non-equilibrium phase transitions, in particular in the
description of self-affine surfaces. We will briefly comment on the
role that this model is meant to play in this physical theory.
Furthermore, we will mention some open questions of physical nature
related to it. Subsequently we will start with the rigorous analysis
of our model, which is a fourth order partial differential equation. We
will describe our progress in building an existence theory for the
full model, which is a parabolic equation, and for its stationary
counterpart. For the latter case existence and multiplicity results
are provided, and for the former one we will show local in time
existence of the solution, that can be made global for small enough
data, and cannot if these are large enough. We will show how these
results fit into the physical theory, and what open questions are left
for the future.