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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Sturmian theorems for second order systems

Author: W. Allegretto
Journal: Proc. Amer. Math. Soc. 94 (1985), 291-296
MSC: Primary 35B05; Secondary 35J45
MathSciNet review: 784181
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Abstract: Sturmian theorem are established for weakly coupled elliptic systems generated in a bounded domain by the expressions $ {l_1}\vec u = - \Delta \vec u + A\vec u,{l_2}\vec w = - \Delta \vec w + B\vec w$, and Dirichlet boundary conditions. Here $ \Delta $ denotes the Laplace operator, and $ A,B$ are $ m \times m$ matrices. We do not assume that $ A,B$ are symmetric, but instead essentially require $ B$ irreducible and $ {b_{ij}} \leqslant 0{\text{ if }}i \ne j$. Estimates on the real eigenvalue of $ {l_2}$, with a positive eigenvector are then obtained as applications. Our results are motivated by recent theorems for ordinary differential equations established by Ahmad, Lazer and Dannan.

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Keywords: Sturmian theorem, elliptic system, eigenvalue, positive operator
Article copyright: © Copyright 1985 American Mathematical Society