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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The steepest point of the boundary layers of singularly perturbed semilinear elliptic problems
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Trans. Amer. Math. Soc. 356 (2004), 2123-2135 Request permission

Abstract:

We consider the nonlinear singularly perturbed problem \[ -\epsilon ^2\Delta u = f(u), \quad u > 0 \quad \mbox {in} \quad \Omega , u = 0 \quad \mbox {on} \quad \partial \Omega , \] 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
  • Received by editor(s): October 3, 2002
  • Received by editor(s) in revised form: July 11, 2003
  • Published electronically: January 6, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 356 (2004), 2123-2135
  • MSC (2000): Primary 35J65, 35J60
  • DOI: https://doi.org/10.1090/S0002-9947-04-03468-3
  • MathSciNet review: 2031056