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7.—Elliptic Singular Perturbations of First-order Operators with Critical Points

Published online by Cambridge University Press:  14 February 2012

P. P. N. de Groen
P. P. N. de Groen
Affiliation:
Wiskundig Seminarium, Vrije Universiteit, Amsterdam
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Synopsis

We study the asymptotic behaviour for ɛ→+0 of the solution Φ of the elliptic boundary value problem

is a bounded domain in ℝ2, 2 is asecond-order uniformly elliptic operator, 1 is a first-order operator, which has critical points in the interior of , i.e. points at which the coefficients of the first derivatives vanish, ɛ and μ are real parameters and h is a smooth function on . We construct firstorder approximations to Φ for all types of nondegenerate critical points of 1 and prove their validity under some restriction on the range of μ.

In a number of cases we get internal layers of nonuniformity (which extend to the boundary in the saddle-point case) near the critical points; this depends on the position of the characteristics of 1 and their direction. At special values of the parameter μ outside the range in which we could prove validity we observe ‘resonance’, a sudden displacement of boundary layers; these points are connected with the spectrum of the operator ɛ2 + 1 subject to boundary conditions of Dirichlet type.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 74 , 1976 , pp. 91 - 113
Copyright
Copyright © Royal Society of Edinburgh 1976

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