Soit un domaine et un paramètre réel positif. Considérons les deux problèmes aux limites sur , et , où et sont des opérateurs différentiels elliptiques et où le degré de est supérieur au degré de .
En utilisant l’interpolation quadratique entre espaces de Hilbert, on étudie les problèmes suivants :
1) Déterminer les normes pour lesquelles converge vers ;
2) Estimer la rapidité de convergence de vers , pour ces normes.
@article{AIF_1968__18_2_135_0, author = {Greenlee, Wilfred M.}, title = {Rate of convergence in singular perturbations}, journal = {Annales de l'Institut Fourier}, pages = {135--191}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {18}, number = {2}, year = {1968}, doi = {10.5802/aif.296}, mrnumber = {39 #3133}, zbl = {0175.40006}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.296/} }
TY - JOUR AU - Greenlee, Wilfred M. TI - Rate of convergence in singular perturbations JO - Annales de l'Institut Fourier PY - 1968 SP - 135 EP - 191 VL - 18 IS - 2 PB - Institut Fourier PP - Grenoble UR - http://www.numdam.org/articles/10.5802/aif.296/ DO - 10.5802/aif.296 LA - en ID - AIF_1968__18_2_135_0 ER -
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. http://www.numdam.org/articles/10.5802/aif.296/
[1] Theory of Bessel potentials, Part. III, In preparation as a Univ. of Kansas Tech. Rep.
, , and ,[2] Theory of Bessel potentials, Part. II, Revised Version, Univ. of Kansas Tech. Rep. 8 (new series), 1964. Ann. Inst. Fourier (Grenoble, 17, 2 (1967)). | Numdam | Zbl
, and ,[3] Lectures on elliptic boundary value problems. Van Nostrand, 1965. | MR | Zbl
,[4] Potentiels Besseliens, Ann. Inst. Fourier (Grenoble) 15 (1965), 43-58. | Numdam | MR | Zbl
,[5] Functional spaces and functiona completion, Ann. Inst. Fourier (Grenoble), 6 (1955-1956), 125-1851. | Numdam | MR | Zbl
and ,[6] Theory of Bessel potentials, I. Ann Inst. Fourier (Grenoble), 11 (1961), 385-475. | Numdam | MR | Zbl
and ,[7] Caractérisation de quelques espaces d'interpolation. Arch. Rational Mech. Anal., 25 (1967), 40-63. | MR | Zbl
,[8] Phénomènes de perturbation singulière dans les problèmes aux limites. Ann. Inst. Fourier (Grenoble), 10 (1960), 61-150. | Numdam | MR | Zbl
,[9] Perturbations singulières. C.R. Acad. Sci. Paris, 259 (1964), 4213-4215. | MR | Zbl
,[10] Tables of functions with formulae and curves, Dover, 1943.
and ,[11] 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). | MR | Zbl
,[12] Quadratic forms in Hilbert spaces and asymptotic perturbation series, Dep't. of Math., Univ. of Calif., Berkeley, (1955).
,[13] On Cauchy's problem for hyperbolic equations and the differentiability of solutions of elliptic equations, Comm. Pure Appl Math., 8 (1955), 615-633. | MR | Zbl
,[14] Parabolic equations, Contributions to the theory of partial differential equations, pp. 167-190, Ann. of Math. Studies, no. 33, Princeton Univ. Press, (1954). | MR | Zbl
and ,[15] Espaces intermédiaires entre espaces hilbertiens et applications, Bull. Math. Soc. Sci. Math. Phys. R.P. Roumaine (N. S.), 2 (50), (1958), 419-432. | MR | Zbl
,[16] Équations différentielles opérationnelles et problèmes aux limites, Springer-Verlag, 1961. | Zbl
,[17] Problemi ai limiti non omogenei, I. Ann. Scuola Norm. Sup. Pisa (3), 14 (1960), 269-308. | Numdam | MR | Zbl
, and ,[18] Problemi ai limiti non omogenei. III. Ann. Scuola Norm. Sup. Pisa (3), 15 (1961), 41-103. | Numdam | Zbl
, and ,[19] Problèmes aux limites non homogènes. IV. Ann. Scuola Norm. Sup. Pisa (3), 15 (1961), 311-326. | Numdam | MR | Zbl
and ,[20] Problèmes aux limites non homogènes, VI, J. Analyse Math., 11 (1963), 165-188. | MR | Zbl
and ,[21] Remarks on strongly elliptic partial differential equations, Comm. Pure Appl. Math., 8 (1955), 649-675. | MR | Zbl
,[22] Elliptic boundary problems with a small parameter, Jour. Math. Anal. Appl., 14 (1966), 341-358. | MR | Zbl
,[1′] Asymptotic phenomena in mathematical physics Bull. Amer. Math. Soc., 61 (1955), 485-504. | MR | Zbl
,[2′] Perturbation theory of semi-bounded operators, Math. Annalen, 125 (1953), 435-447. | MR | Zbl
,[3′] Perturbation theory for linear operators, Springer-Verlag, 1966. | MR | Zbl
,[4′] Singular perturbation of eigenvalue problems for linear differential equations of even order, Comm. Pure and Appl. Math., 8 (1955), 251-278. | MR | Zbl
,[5′] Theory of Sound, Vol. I and II, Dover, 1945. | MR | Zbl
, Lord,[6′] Singular perturbations of non-linear elliptic and parabolic variational boundary-value problems, Can. J. Math., 18 (1966), 861-872. | MR | Zbl
,[7′] Singular perturbations and non-linear parabolic boundary value problems, Jour. Math. Anal. Appl., 17 (1967), 248-261. | MR | Zbl
,[8′] 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
and ,[9′] Asymptotic expansions for ordinary differential equations, (Interscience), Wiley, 1966.
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