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Paper
IPM / M / 15166 |
School of Mathematics
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Title: |
Spherical twists as the Σ2-harmonic maps from N-dimensional annuli into \SpN−1
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Author(s): |
Mohammad Sadegh Shahrokhi-Dehkordi
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Status: |
Published
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Journal: |
Potential Anal.
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Year: |
2018
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Pages: |
DOI: 10.1007/s11118-018-9684-8
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Supported by: |
IPM
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Abstract: |
Let \X ⊂ \Rn be a bounded Lipschitz domain and consider the σ2-energy functional
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\mathbb Fσ2[u; \X] : = | ⌠
⌡
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\X
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⎢
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∧2 ∇u | ⎢
⎢
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2
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dx, |
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over the space of admissible Sobolev maps
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A(\X) := | ⎧
⎨
⎩
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u ∈ W1,4(\X, \Spn−1) : u|∂\X = x|x|−1 | ⎫
⎬
⎭
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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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