Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions (Q424422)
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scientific article; zbMATH DE number 6040258
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions |
scientific article; zbMATH DE number 6040258 |
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Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions (English)
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1 June 2012
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In this paper, the author proves even symmetry and monotonicity of certain solutions of the Allen-Cahn equation \(u_{xx}+u_{yy}-F'(u)=0\) in a half plane. The author also shows that entire solutions with finite Morse index and four ends must be evenly symmetric with respect to two orthogonal axes. A classification scheme of general entire solutions with finite Morse index is also presented using energy quantization.
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Allen-Cahn equation
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Hamiltonian identity
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level set
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saddle solutions
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even symmetry
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monotonicity
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Morse index
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