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

Abstract

Given a continuous mapping φ:M→N of compact manifolds we prove that there exists a differentiable solution of the problem E_p(u):=∫_M|du|^pdv $\leadsto$ Min in the homotopy class [φ] provided the sectional curvature of N is nonpositive. Here p∈[2,∞) 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<ε<1) E_p^ε(u):=∫_M(ε+|du|²)^p/2dv $\leadsto$ min in [φ] for which we establish the existence of solutions u_ε 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.