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interior transition layer

Proceedings of the Royal Society of Edinburgh Section A: Mathematics (1)

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Locations of interior transition layers to inhomogeneous transition problems in higher -dimensional domains
Part of
- Partial differential equations
- Elliptic equations and systems
Zhuoran Du
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 153 / Issue 3 / June 2023

Published online by Cambridge University Press:

08 March 2022, pp. 764-783

Print publication:

June 2023
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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.