Nonlinear Problems in Field Theories

There are many interesting and challenging problems in the area of classical eld theories. This area has attracted the attention of algebraists, geometers, and topol-ogists in the past and has begun to attract more analysts. Analytically, the area ooers all types of diierential equation problems which come from the two basic sets of equations in physics describing fundamental interactions, namely, the Yang{Mills equations governing electromagnetic, weak, and strong forces, reeecting internal symmetry , and the Einstein equations governing gravity, reeecting external symmetry. Of course, a combination of these two sets of equations gives us a theory which couples both symmetries and uniies all forces. In these lectures, I will present some of the problems that I have been interested in which involve elliptic equations. 1 Self-dual cosmic strings Cosmic strings arise as nite-energy static solutions of the coupled Einstein and Yang{ Mills{Higgs equations in which the gravitational metric housed over a Riemann surface M and eld conngurations are independent of time and a third vertical variable. Such a structure leads to energy and curvature concentrations at the centers of strings, around which matter accretion takes place, which provides a mechanism for galaxy formation. Mathematically, Abelian self-dual strings are well understood. Let us begin with the Abelian Higgs model for which the energy density and momentum tensor are given by