As is well known, many phenomena in Engineering and Mathematical Physics can be
modeled by means of boundary value problems for a certain elliptic differential operator L in a domain 
omega.
When L is a differential operator of second order, a variety of tools are available for dealing
with such problems including boundary integral methods, variational methods, harmonic
measure techniques, and methods based on classical harmonic analysis. The situation when
the differential operator has higher order (as is the case for instance with anisotropic plate
bending when one deals with fourth order) stands in sharp contrast with this as only fewer
option could be successfully implemented. Alberto Calder´on, one of the founders of the
modern theory of Singular Integral Operators, has advocated in the seventies the use of
layer potentials for the treatment of higher order elliptic boundary value problems. While
the layer potential method has proved to be tremendously successful in the treatment of
second order problems, this approach is currently insufficiently developed to deal with the
intricacies of the theory of higher order operators. In fact, it is largely absent from the
literature dealing with such problems.
In this talk I will discuss recent progress in developing a multiple layer potential approach
for the treatment of boundary value problems associated with higher order elliptic differential
operators. This is done in a very general class of domains which is in the nature of best
possible from the point of view of geometric measure theory.