Solutions to the Allen Cahn Equation and Minimal Surfaces
Publication in refereed journal


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摘要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 Delta u + u - u(3) = 0. In particular, we consider a minimal surface G in 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 R-9 which is monotone in the x(9) direction whose zero level set lies close to a large dilation of Gamma, 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.
著者del Pino M, Wei JC
期刊名稱Milan Journal of Mathematics
出版年份2011
月份6
日期1
卷號79
期次1
出版社Springer Verlag (Germany)
頁次39 - 65
國際標準期刊號1424-9286
電子國際標準期刊號1424-9294
語言英式英語
關鍵詞Infinite dimensional Lyapunov-Schmidt reduction; Jacobi operator; Minimal surfaces
Web of Science 學科類別Mathematics; MATHEMATICS; Mathematics, Applied; MATHEMATICS, APPLIED

上次更新時間 2021-19-02 於 23:32