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Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains

Part of: Elliptic equations and systems Partial differential equations

Published online by Cambridge University Press:  08 March 2022

Zhuoran Du
Zhuoran Du*
Affiliation:
School of Mathematics, Hunan University, Changsha 410082, PR China (duzr@hnu.edu.cn)
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Abstract

We consider the following inhomogeneous problems

\[ \begin{cases} \epsilon^{2}\mbox{div}(a(x)\nabla u(x))+f(x,u)=0 & \text{ in }\Omega,\\ \frac{\partial u}{\partial \nu}=0 & \text{ on }\partial \Omega,\\ \end{cases} \]
where $\Omega$ is a smooth and bounded domain in general dimensional space $\mathbb {R}^{N}$, $\epsilon >0$ is a small parameter and function $a$ is positive. We respectively obtain the locations of interior transition layers of the solutions of the above transition problems that are $L^{1}$-local minimizer and global minimizer of the associated energy functional.

Keywords

Local minimizerinterior transition layerinterface locationinhomogeneous transition problems

MSC classification

Primary: 35B25: Singular perturbations
Secondary: 35J20: Variational methods for second-order elliptic equations 35J61: Semilinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 3 , June 2023 , pp. 764 - 783
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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