Solutions to the Allen Cahn Equation and Minimal Surfaces |
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Authors: | Manuel del Pino Juncheng Wei |
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Institution: | 1.Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS),Universidad de Chile,Santiago,Chile;2.Department of Mathematics,Chinese University of Hong Kong,Shatin,Hong Kong |
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Abstract: | We discuss and outline proofs of some recent results on application of singular perturbation techniques for solutions in entire
space of the Allen-Cahn equation Δu + u − u
3 = 0. In particular, we consider a minimal surface Γ in
\mathbb R9{\mathbb {R}^9} which is the graph of a nonlinear entire function x
9 = F(x
1, . . . , x
8), found by Bombieri, De Giorgi and Giusti, the BDG surface. We sketch a construction of a solution to the Allen Cahn equation
in
\mathbb R9{\mathbb {R}^9} which is monotone in the x9 direction whose zero level set lies close to a large dilation of Γ, recently obtained by M. Kowalczyk and the authors. This
answers a long standing question by De Giorgi in large dimensions (1978), whether a bounded solution should have planar level
sets. We sketch two more applications of the BDG surface to related questions, respectively in overdetermined problems and
in eternal solutions to the flow by mean curvature for graphs. |
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Keywords: | |
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