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

Article type: Research Article

Authors: Duzaar, Frank | Fuchs, Martin

Affiliations: Institut für Angewandte Math. der Univ. Bonn, SFB 256, Beringstr. 4, Universitát Bonn, D-5300 Bonn 1, Germany | TH Darmstadt Fachbereich 4/Ag 6, Schloßgartenstr. 7, D-6100 Darmstadt, Germany

Abstract: Given a continuous mapping φ:M→N of compact manifolds we prove that there exists a differentiable solution of the problem Ep(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) Epε(u):=∫M(ε+|du|2)p/2dv$\leadsto$min in [φ] for which we establish the existence of solutions uε 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.

DOI: 10.3233/ASY-1991-5203

Journal: Asymptotic Analysis, vol. 5, no. 2, pp. 129-144, 1991