Singularly perturbed semilinear elliptic boundary value problems
Singularly perturbed semilinear elliptic boundary value problems
复制标题
奇摄动半线性椭圆边值问题
DOI：
10.1080/03605307908820090
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发表时间：
1979
影响因子：
1.9
通讯作者：
F. Howes
中科院分区：
文献类型：
--
作者：
F. Howes
1.导论.对于E（a）E~ u+ Lu= r（x，o）2 1，u在En上规定的小正值，在区域ncR（N2 2）中的奇摄动线性Dirichlet问题N，其中L2是一致椭圆二阶微分算子，L1是一阶微分算子，特别是在N= 2的情况下，已有许多作者考虑过（参见，例如，[331，[261，[321，[281，[10]，[1191，[111，[E1，[201，[251）。由于问题（d）是线性的，因此在对算子L1和L2的系数以及非齐次项r的温和限制下，保证了E> 0的每个值的解的存在性。因此，有趣而重要的问题是描述当E趋于零时，（4）的解如何表现。不幸的是，当区域n足够复杂时，如当算子L1的特征与C1的边界的部分重合或当L1的系数在~ 7中具有临界点时，在该线性理论中已经存在许多困难（参见，例如，[10]、[191]、[31]、[41]、[81]、[F71]、[251.）。考虑到与线性问题有关的困难，Marcel Dekker，Inc. All Rights Reserved.未经出版商书面许可，不得以任何形式或通过任何电子或机械手段（包括影印、缩微、MD记录）或任何信息存储和检索系统复制或传播本作品或其任何部分。
1. Introduction. The singularly perturbed linear Dirichlet problem N in a region ncR (N 2 2) for small positive values of E (a) E~ u+ Lu= r (x, o) 2 1, u prescribedon En, where L2 is a uniformly elliptic second-order differential operator and L1 is a first-order differential operator has been considered by a number of authors especially in the case N= 2 (cf., for example,[331,[261,[321,[281,[lo], 1191,[ill,[E l,[201,[251). Since the problem (d) is linear the existence of a solution for each value of E> 0 is guaranteed under mild restrictions on the coefficients of the operators L1 and L2 and on the nonhomogeneous term r. Thus the interesting and important problem consists of describing how solutions of (4 behave as E tends to zero. Uhfortunately many difficulties are already present in this linear theory when the region n is sufficiently complicated as is the case when the characteristics of the operator L1 coincide with portions of the boundary of Cl or when the coefficients of L1 have critical points in~ 7 (cf., for example,[lo],[191,[31,[41,[81, F71,[251.). In view of the difficulties associated with the linear problem it is not surprising that the nonlinear problemCopyaht@ 1979 by Marcel Dekker, Inc. All Rights Reserved. Neither this work nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, includ~ ng photocopying, microfilming, md recording, or by any information storage and retrieval system, without permission in writing from the publisher.