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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems


Author: T. Shibata
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 2123-2135
MSC (2000): Primary 35J65, 35J60
Published electronically: January 6, 2004
MathSciNet review: 2031056
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Abstract: We consider the nonlinear singularly perturbed problem

\begin{displaymath}-\epsilon^2\Delta u = f(u), \enskip u > 0 \quad \mbox{in} \enskip \Omega, u = 0 \quad \mbox{on} \enskip \partial\Omega, \end{displaymath}

where $\Omega \subset {\mathbf{R}}^N$ ($N \ge 2$) is an appropriately smooth bounded domain and $\epsilon > 0$ is a small parameter. It is known that under some conditions on $f$, the solution $u_\epsilon$ corresponding to $\epsilon$ develops boundary layers when $\epsilon \to 0$. We determine the steepest point of the boundary layers on the boundary by establishing an asymptotic formula for the slope of the boundary layers with exact second term.


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Additional Information

T. Shibata
Affiliation: The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima, 739-8521, Japan
Email: shibata@mis.hiroshima-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03468-3
PII: S 0002-9947(04)03468-3
Keywords: Boundary layer, singular perturbation, semilinear elliptic equations
Received by editor(s): October 3, 2002
Received by editor(s) in revised form: July 11, 2003
Published electronically: January 6, 2004
Article copyright: © Copyright 2004 American Mathematical Society