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Singular Perturbations of Non-Linear Elliptic and Parabolic Variational Boundary-Value Problems

Published online by Cambridge University Press:  20 November 2018

Bui An Ton
Bui An Ton*
Affiliation:
University of Montreal
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Singular perturbations of linear elliptic and parabolic boundary-value problems have been studied extensively by Visik and Lyusternik (7), Huet (5), and others. It is the purpose of this paper to extend the results of (5) to the non-linear elliptic and parabolic variational boundary-value problems considered during the last few years by Browder (2, 4).

In §1, we give the notations and state the main assumptions on the nonlinearity of the elliptic operators. In §2 we study the singular perturbations of non-linear elliptic variational boundary problems. In §3, we consider the case of non-linear parabolic variational boundary problems with a small parameter.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 861 - 872
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Aubin, J. P., Un théorème de compacité, C. R. Acad. Sci., Paris, 256 (1963), 5042–5044.Google Scholar
2. Browder, F. E., Variational boundary value problems for quasi linear elliptic equations of arbitrary order, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 31–37.Google Scholar
3. Browder, F. E., Variational boundary value problems for quasi linear elliptic equations II, Proc. Nat. Acad. Sci. U.S.A., 50 (1963), 592–598.Google Scholar
4. Browder, F. E., Strongly non-linear parabolic boundary value problems, Amer. J. Math., 86 (1964), 339–357.Google Scholar
5. Huet, D., Phénomènes de perturbations singulières, Ann. Inst. Fourier, Grenoble, 10 (1960), 1–96.Google Scholar
6. Ton, B. A., Elliptic boundary problems with a small parameter, J. Math. Anal. Appl., 4 (1966).Google Scholar
7. Visik, M. I. and Luysternik, L. A., Regular degeneration and boundary layer for linear differential equations with a small parameter, Uspehi Mat. Nauk (N.S.), 12 (1957), 3–122 = Amer. Math. Soc. Transi. Ser. 2, 20 (1962), 239-364.Google Scholar