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Symmetry of some entire solutions to the Allen-Cahn equation in two dimensions. (English) Zbl 1250.35078
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.

MSC:
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
49Q05 Minimal surfaces and optimization
53A04 Curves in Euclidean and related spaces
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References:
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