On the existence and uniqueness of exponentially harmonic maps and some related problems

Abstract

The family of partial differential equations −eΔu − 2Δ∞u = 0 (e > 0) is studied in a bounded domain Ω for given boundary data. In the case where e = 1, which is closely related to the study of exponentially harmonic maps, we establish existence and uniqueness of a classical solution as the unique minimizer in a closed subset of an Orlicz–Sobolev space of the appropriate energy functional associated to this problem—the integral over Ω of the exponential energy density $$u \mapsto {1 \over 2}\exp \left( {{{\left| {\nabla u} \right|}^2}} \right)$$ . We also explore the connections between the classical solutions of these problems and infinity harmonic and harmonic maps by studying the limiting behavior of the solutions as e → 0+ and e → ∞, respectively. In the former case, we recover a result of Evans and Yu [6].