Existence and regularity of functions which minimize certain energies in homotopy classes of mappings

Abstract

Duzaar. F. and M. Fuchs, Existence and regularity of functions which minimize certain energies in homotopy classes of mappings, Asymptotic Analysis 5 (1991) 129-144. Given a continuous mapping 'P: M --> N of compact manifolds we prove that there exists a differentiable solution of the problem Ep(u) ,= fM I du I p dv -> Min in the homotopy class ['1'1 provided the sectional curvature of N is nonpositive. Here p E [2, (0) is an arbitrary real number, hence we obtain an extension of a well-known theorem of Eells and Sampson. Our arguments are based on an asymptotic analysis of the less degenerate problems (0 min in ['1'1 for which we establish the existence of solutions u, which converge to a minimizer of our original variational problem. As further applications the paper contains various estimates for p-harmonic maps into negatively curved target manifolds.