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VOL. 8 | 1984 A semilinear elliptic boundary-value problem describing small patches of vorticity in an otherwise irrotational flow
Grant Keady

Editor(s) Neil S. Trudinger, Graham H. Williams

Abstract

Let $\Omega$ be a bounded domain in $\mathbb{R}^2$. The study, begun in Keady[l981] and Keady and Kloeden[1984] of the boundary-value problem, for (\lambda/k, \psi) \[-\Delta \psi \in \lambda H (\psi - k) \hspace{.1in} in \hspace{.1in} \Omega \subset \mathbb{R}^2, \psi = 0 \hspace{.1in} on \hspace{.1in} \partial \Omega, \] is continued. Here $\Delta$ denotes the Laplacian, $H$ is the Heaviside step function and one of $\lambda$ or $k$ is a given positive constant. The solutions considered always have $\psi \gt 0$ in $\Omega$ and $\lambda/k \gt 0$, and have cores \[ A = {(x,y) \in \Omega | \psi(x,y) \gt k} \] In the special case $\Omega = B(O,R)$ , a disc, the explicit exact solutions are available. They satisfy \[ (\ast) \hspace{2in} (\psi_{m} - k)/k \rightarrow 0 \hspace{.1in} as \hspace{.1in} area(A) \rightarrow 0 , \] where $\psi_m$ is the maximum of $\psi$ over $\Omega$. Here (\ast) will be established for other domains. An adaptation of the maximum principles of Gidas, Ni and Nirenberg [1979] is an important step in establishing the above result.

Information

Published: 1 January 1984
First available in Project Euclid: 18 November 2014

zbMATH: 0568.35040
MathSciNet: MR799219

Rights: Copyright © 1984, Centre for Mathematical Analysis, The Australian National University. This book is copyright. Apart from any fair dealing for the purpose of private study, research, criticism or review as permitted under the Copyright Act, no part may be reproduced by any process without permission. Inquiries should be made to the publisher.

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Vol. 8 • 1 January 1984
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