no. 2
Rate of convergence in singular perturbations
Annales de l'Institut Fourier, Tome 18 (1968) no. 2, pp. 135-191

Soit D⊂R n un domaine et ε un paramètre réel positif. Considérons les deux problèmes aux limites sur D, (ε𝒰+ℬw ε =f et ℬu=f, où 𝒰 et ℬ sont des opérateurs différentiels elliptiques et où le degré de 𝒰 est supérieur au degré de ℬ.

En utilisant l’interpolation quadratique entre espaces de Hilbert, on étudie les problèmes suivants :

1) Déterminer les normes pour lesquelles w ε converge vers u ;

2) Estimer la rapidité de convergence de w ε vers u, pour ces normes.

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     author = {Greenlee, Wilfred M.},
     title = {Rate of convergence in singular perturbations},
     journal = {Annales de l'Institut Fourier},
     pages = {135--191},
     year = {1968},
     publisher = {Institut Fourier},
     address = {Grenoble},
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Greenlee, Wilfred M. Rate of convergence in singular perturbations. Annales de l'Institut Fourier, Tome 18 (1968) no. 2, pp. 135-191. doi: 10.5802/aif.296

[1] R. Adams, N. Aronszajn, and M. Hanna, Theory of Bessel potentials, Part. III, In preparation as a Univ. of Kansas Tech. Rep.

[2] R. Adams, N. Aronszajn and K. T. Smith, Theory of Bessel potentials, Part. II, Revised Version, Univ. of Kansas Tech. Rep. 8 (new series), 1964. Ann. Inst. Fourier (Grenoble, 17, 2 (1967)). | Zbl | Numdam

[3] S. Agmon, Lectures on elliptic boundary value problems. Van Nostrand, 1965. | Zbl | MR

[4] N. Aronszajn, Potentiels Besseliens, Ann. Inst. Fourier (Grenoble) 15 (1965), 43-58. | Zbl | MR | Numdam

[5] N. Aronszajn and K. T. Smith, Functional spaces and functiona completion, Ann. Inst. Fourier (Grenoble), 6 (1955-1956), 125-1851. | Zbl | MR | Numdam

[6] N. Aronszajn and K. T. Smith, Theory of Bessel potentials, I. Ann Inst. Fourier (Grenoble), 11 (1961), 385-475. | Zbl | MR | Numdam

[7] P. Grisvard, Caractérisation de quelques espaces d'interpolation. Arch. Rational Mech. Anal., 25 (1967), 40-63. | Zbl | MR

[8] D. Huet, Phénomènes de perturbation singulière dans les problèmes aux limites. Ann. Inst. Fourier (Grenoble), 10 (1960), 61-150. | Zbl | MR | Numdam

[9] D. Huet, Perturbations singulières. C.R. Acad. Sci. Paris, 259 (1964), 4213-4215. | Zbl | MR

[10] E. Jahnke and F. Emde, Tables of functions with formulae and curves, Dover, 1943.

[11] J. Kadlec, The regularity of the solution of the Poisson problem in a domain whose boundary is similar to that of a convex domain, Czechoslovak Math. J., 14 (89) (1964), 386-393 (In Russian). | Zbl | MR

[12] T. Kato, Quadratic forms in Hilbert spaces and asymptotic perturbation series, Dep't. of Math., Univ. of Calif., Berkeley, (1955).

[13] P. D. Lax, On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl Math., 8 (1955), 615-633. | Zbl | MR

[14] P. D. Lax and A. N. Milgram, Parabolic equations, Contributions to the theory of partial differential equations, pp. 167-190, Ann. of Math. Studies, no. 33, Princeton Univ. Press, (1954). | Zbl | MR

[15] J. L. Lions, Espaces intermédiaires entre espaces hilbertiens et applications, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumaine (N. S.), 2 (50), (1958), 419-432. | Zbl | MR

[16] J. L. Lions, Équations différentielles opérationnelles et problèmes aux limites, Springer-Verlag, 1961. | Zbl

[17] J. L. Lions, and E. Magenes, Problemi ai limiti non omogenei, I. Ann. Scuola Norm. Sup. Pisa (3), 14 (1960), 269-308. | Zbl | MR | Numdam

[18] J. L. Lions, and E. Magenes, Problemi ai limiti non omogenei. III. Ann. Scuola Norm. Sup. Pisa (3), 15 (1961), 41-103. | Zbl | Numdam

[19] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes. IV. Ann. Scuola Norm. Sup. Pisa (3), 15 (1961), 311-326. | Zbl | MR | Numdam

[20] J. L. Lions and E. Magenes, Problèmes aux limites non homogènes, VI, J. Analyse Math., 11 (1963), 165-188. | Zbl | MR

[21] L. Nirenberg, Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675. | Zbl | MR

[22] B. A. Ton, Elliptic boundary problems with a small parameter, Jour. Math. Anal. Appl., 14 (1966), 341-358. | Zbl | MR

[1′] K. Friedrichs, Asymptotic phenomena in mathematical physics Bull. Amer. Math. Soc., 61 (1955), 485-504. | Zbl | MR

[2′] T. Kato, Perturbation theory of semi-bounded operators, Math. Annalen, 125 (1953), 435-447. | Zbl | MR

[3′] T. Kato, Perturbation theory for linear operators, Springer-Verlag, 1966. | Zbl | MR

[4′] J. Moser, Singular perturbation of eigenvalue problems for linear differential equations of even order, Comm. Pure and Appl. Math., 8 (1955), 251-278. | Zbl | MR

[5′] Rayleigh, Lord, Theory of Sound, Vol. I and II, Dover, 1945. | Zbl | MR

[6′] B. A. Ton, Singular perturbations of non-linear elliptic and parabolic variational boundary-value problems, Can. J. Math., 18 (1966), 861-872. | Zbl | MR

[7′] B. A. Ton, Singular perturbations and non-linear parabolic boundary value problems, Jour. Math. Anal. Appl., 17 (1967), 248-261. | Zbl | MR

[8′] M. I. Višik and L. A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small paramater, Uspehi Mat. Nauk. (N.S.), 12 (1957), no. 5 (77), 3-122 ; Am. Math. Soc. Trans. Ser., 2, 20 (1962), 239-364. | Zbl

[9′] W. Wasow, Asymptotic expansions for ordinary differential equations, (Interscience), Wiley, 1966.

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