Heteroclinic solutions of singular quasilinear bistable equations

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• 作者：Denis Bonheure ; Isabel Coelho ; Manon Nys
• 关键词：Mean curvature operator in Lorentz–Minkowski space ; Free energy functional ; Phase transition ; Increasing rearrangement ; Rigidity ; Symmetry
• 刊名：Nonlinear Differential Equations and Applications NoDEA
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：24
• 期：1
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1420-9004
• 卷排序：24

文摘

In this note we consider the action functional $$\begin{aligned} \int _{\mathbb {R}\times \omega } \left( 1 - \sqrt{1 - |\nabla u|^{2}} + W(u) \right) \mathrm {d}\bar{x} \end{aligned}$$where W is a double well potential and \(\omega \) is a bounded domain of \(\mathbb {R}^{N-1}\). We prove existence, one-dimensionality and uniqueness (up to translations) of a smooth minimizing phase transition between the two stable states \(u = -1\) and \(u = 1\). The question of existence of at least one minimal heteroclinic connection for the non-autonomous model $$\begin{aligned} \int _{\mathbb {R}} \left( 1 - \sqrt{1 - |u'|^{2}} + a(t)W(u) \right) \mathrm {d}t \end{aligned}$$is also addressed. For this functional, we look for the possible assumptions on a(t) ensuring the existence of a minimizer.KeywordsMean curvature operator in Lorentz–Minkowski spaceFree energy functionalPhase transitionIncreasing rearrangementRigiditySymmetryD.B. is supported by INRIA-Team MEPHYSTO, MIS F.4508.14 (FNRS), PDR T.1110.14F (FNRS) & ARC AUWB-2012-12/17-ULB1- IAPAS. I.C. gratefully acknowledges the support of Fundação para a Ciência e a Tecnologia, through the scholarship SFRH/BD/61484/2009, during her stay at the Université Libre de Bruxelles. M.N. is supported by the project ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems-COMPAT”. M.N. was supported by the FNRS when this research was initiated.