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

  • Marian Bocea
  • Mihai MihăilescuEmail author
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Abstract

The family of partial differential equations −εΔu − 2Δu = 0 (ε > 0) is studied in a bounded domain Ω for given boundary data. In the case where ε = 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 ε → 0+ and ε → ∞, respectively. In the former case, we recover a result of Evans and Yu [6].

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© Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsLoyola University ChicagoChicagoUSA
  2. 2.Department of MathematicsUniversity of CraiovaCraiovaRomania
  3. 3.“Simion Stoilow” Institute of Mathematics of the Romanian AcademyBucharestRomania

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