Partial regularity of harmonic maps to spheres

Abstract

Let n,k be integers greater than or equal to 2 and omega a bounded open subset of the Euclidean space R(n). A (minimizing) harmonic map to sphere u:omega -> S(k-1) is a function in the Sobolev space H1(omega,S(k-1)) which satisfies the following system of nonlinear partial differential equation -delta = u/delta u/2, and u minimizes the fnctional E(u) = 1/2 the integral of omega /delta u/2 dx among those w E H1 (omega,S(k-1)) such that u is equivalent to w in a neighborhood of omega. The key theorem for distingushing the singular and the regular points of harmonic maps is the small energy regularity theorem. In this thesis we give an alternative proof for this theorem when the target spaces are spheres via the penalty approximation technique.