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- Title
Coexistence of two interfaces for an anisotropic Fife-Greenlee equation.
- Authors
Liang, Wenyin; Yang, Jun
- Abstract
We consider the nonlinear problem of anisotropic Fife-Greenlee equation $ \varepsilon^2{\mathrm {div}}\big(\nabla_{{\mathfrak a}(y)} u\big)+\big(u-{\mathcal P}(y)\big)(1-u^2) = 0\quad \mbox{in}\ \Omega, \qquad \nabla_{{\mathfrak a}(y)} u\cdot \nu = 0\quad \mbox{on}\ \partial \Omega, $ where $ \Omega $ is a bounded domain in $ {\mathbb R}^2 $ with smooth boundary, $ \varepsilon $ is a small positive parameter, $ \nu $ denotes the unit outward normal of $ \partial\Omega $, and $ {\mathcal P}(y) $ is a smooth function on $ \bar{\Omega} $. The operator $ \nabla_{{\mathfrak a}(y)} u $ is defined by$ \nabla_{{\mathfrak a}(y)} u = \big({\mathfrak a}_1(y)u_{y_1}, {\mathfrak a}_2(y)u_{y_2}\big) \quad \mbox{with }\ {\mathfrak a}(y) = \big({\mathfrak a}_1(y), {\mathfrak a}_2(y)\big), $where $ {\mathfrak a}_1(y) $ and $ {\mathfrak a}_2(y) $ are two positive smooth functions on $ \bar\Omega $.Let $ \Gamma = \{y \in\Omega:{\mathcal P}(y) = 0\} $ be a simple smooth curve in $ \Omega $ that intersects $ \partial\Omega $ orthogonally at exactly two points and divide the domain $ \Omega $ into two parts $ \Omega_- = \{y\in \Omega: {\mathcal P}(y)>0\} $ and $ \Omega_+ = \{y\in \Omega: {\mathcal P}(y)<0\} $. In addition, $ \Upsilon = \{y \in\Omega_+:{\mathcal P}(y) = -3\} $ is a simple smooth curve that intersects $ \partial\Omega $ orthogonally at exactly two points and divide the domain $ \Omega_+ $ into two parts $ \Omega_{+, 1} = \{y\in \Omega_+: 0>{\mathcal P}(y)>-3\} $ and $ \Omega_{+, 2} = \{y\in \Omega_+: -3>{\mathcal P}(y)>-4\} $. By assuming some additional constraints on the functions $ {\mathfrak a}(y) $, $ {\mathcal P}(y) $ as well as the curves $ \Gamma $, $ \Upsilon $ and $ \partial\Omega $, we construct a solution with coexistence of two interfaces approaching $ \Gamma $ and $ \Upsilon $ such that: as $ \varepsilon\rightarrow 0 $,$ u_{\varepsilon}\rightarrow -1\ \mbox{in}\ \Omega_-, \qquad u_{\varepsilon}\rightarrow +1\ \mbox{in}\ \Omega_{+, 1}, \qquad u_{\varepsilon}\rightarrow -3\ \mbox{in}\ \Omega_{+, 2}. $Some other solutions with interfaces will also be constructed in a similar way.
- Subjects
SMOOTHNESS of functions; NONLINEAR equations; EQUATIONS
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2024, Vol 44, Issue 7, p1
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2024011