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Paper   IPM / M / 15166
School of Mathematics
  Title:   Spherical twists as the Σ2-harmonic maps from N-dimensional annuli into \SpN−1
  Author(s):  Mohammad Sadegh Shahrokhi-Dehkordi
  Status:   Published
  Journal: Potential Anal.
  Year:  2018
  Pages:   DOI: 10.1007/s11118-018-9684-8
  Supported by:  IPM
Let \X ⊂ \Rn be a bounded Lipschitz domain and consider the σ2-energy functional
\mathbb Fσ2[u; \X] : =


over the space of admissible Sobolev maps
A(\X) :=

uW1,4(\X, \Spn−1) : u|∂\X = x|x|−1

In this article we address the question of multiplicity versus uniqueness for extremals and strong local minimizers of the σ2-energy funcional \mathbb Fσ2[·, \X] in A(\X) where the domain \mathbb X is n-dimensional annuli. We consider a topological class of maps referred to as spherical twists and examine them in connection with the Euler-Lagrange equations associated with σ2-energy functional over A(\X), the so-called σ2-harmonic map equation on \X. The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations.

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