On rapidly oscillating solutions of a nonlinear elliptic equation
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Houssem Eddine Kadem
Abstract
The aim of this article is to examine the solutions of the boundary value problem of the nonlinear elliptic equation ε2△u = f(u). We describe the asymptotic behavior as ε tends to zero of the solutions on a spherical crown C of RN, (N ≥ 2) in a direct non-classical formulation which suggests easy proofs. We propose to look for interesting solutions in the case where the condition at the edge of the crown is a constant function. Our results are formulated in classical mathematics.Their proofs use the stroboscopic method which is a tool of the nonstandard asymptotic theory of differential equations.
Communicated by Alberto Lastra
References
[1] Bendaas, S.—Alaa, N.: Periodic wave shock solutions of Burgers equations, A new approach, Int. J. Nonlinear Anal. Appl. 10(1) (2019), 119–129.10.1080/25742558.2018.1463597Suche in Google Scholar
[2] Bendaas, S.: Confluence of shocks in Burgers equation. A new approach, Int. J. Differ. Equ. 14(N4) (2015), 369–382.Suche in Google Scholar
[3] Bendaas, S.: Boundary value problems for Burgers equations through nonstandard analysis, Appl. Math. N6 (2015), 1086–1098.10.4236/am.2015.66099Suche in Google Scholar
[4] Bendaas, S.: Léquation de Burgers avec un terme dissipatif. Une approche non standard, An. Univ. Oradea Fasc. Mat. 15 (2008), 239–252.Suche in Google Scholar
[5] Bendaas, S.: Quelques Applications de l’A.N.S aux E.D.P, Thèse de Doctorat. Université de Haute Alsace, France, 1994.Suche in Google Scholar
[6] Callot, J. L.: Stroboscopie infinitésimale, (Strasbourg-Obernai, 1995), 95–124. Prépubl. Inst. Rech. Math. Av., 1995/13, Univ.Louis Pasteur, Strasbourg, 1995.Suche in Google Scholar
[7] Callot, J. L.—Sari, T.: Stroboscopie infinitésimale et moyennisation dans les systèmes déquations différentielles à solutions rapidement oscillantes. In: Outils et modèles mathématiques pour l’automatique, l’analyse des systèmes et le traitement du signal (I. D. Landau, ed.), vol. 3, Editions du CNRS, 1983, pp. 345–353.Suche in Google Scholar
[8] Carrier, G. F.—Pearson, C. E.: Ordinary Differential Equations, Blaisdell Waltham, 1968.Suche in Google Scholar
[9] De Jager, E. M.: Singular Perturbations Problems for linear Differential Equations of Elliptic Type, Arch. Rat. Mech. Anal. 23 (1966).10.1007/BF00281135Suche in Google Scholar
[10] Diener, M.—Lobry, C.: Actes de l’Ecole d’Eté: Analyse Non Standard et Représentation du Réel, Proceeding of the Summer School on Nonstandard Analysis and Representation of the Real, Office des Publications Universitaires, Alger, 1985.Suche in Google Scholar
[11] Diener, M.—Lobry, C.: Nonstandard Analysis in Practice, Chapters 4 and 10, Universitext, Springer Verlag, 1995.10.1007/978-3-642-57758-1Suche in Google Scholar
[12] Fife, P. C.—Greenlee, W. M.: Interior Transition Layers for Elliptic Boundary Value Problems with a Small parameter, Russian Math. Surveys 29(4) (1974), 130.10.1070/RM1974v029n04ABEH001291Suche in Google Scholar
[13] Lakrib, M.—Sari, T.: Time averaging for ordinary differental equations and retard functional differential equations, Electron. J. Differential Equations 2010(40) (2010), http://ejde.math.txstate.edu or http://ejde.math.unt.edu.Suche in Google Scholar
[14] Lutz, R.—Sari, T.: Application of Non Standard Analysis to Boundary Value Problems in Singular Perturbations Theory, Proceeding Oberwolfach 1981, edited by W. Eckhaus and E. M. Dejager, Lectures Notes in Math. 942, Springer Verlag Berlin, 1982, pp. 113–135.10.1007/BFb0094743Suche in Google Scholar
[15] Lutz, R.—Goze, M.: Nonstandard Analysis: A Practical Guide with Applications, Lecture Notes in Math. 881, Springer-Verlag, Berlin, 1981.10.1007/BFb0093397Suche in Google Scholar
[16] Malley, R. E. O.: Phase-plane solutions to some singular perturbation problem, J. Math. Anal. Appl. 54 (1976), 449–466.10.1016/0022-247X(76)90214-6Suche in Google Scholar
[17] Nelson, E.: Internal Set Theory: a new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83(6) (1977), 1165–1198.10.1090/S0002-9904-1977-14398-XSuche in Google Scholar
[18] Nouri, Z.—Bendaas, S.—Kadem, H. E.: N wave and periodic wave solutions for Burgers equations, Int. J. Anal. Appl. 18(2) (2020), 304–318.Suche in Google Scholar
[19] Reeb, G.: Equations Différentielles et Analyse Non Standard, Proceeding of the IV-Int Colloque on Differential Geometry, Santiago de Compostella, 1978.Suche in Google Scholar
[20] Robinson, A.: Nonstandard Analysis, American Elsevier, New York, 1974.Suche in Google Scholar
[21] Sari, T.: Moyennisation Dans les Systè mes Différentiels à Solutions Rapidement Oscillantes, Thèse de Doctorat, Mulhouse, 1983.Suche in Google Scholar
[22] Sari, T.: Nonstandard perturbation theory of differential equations, presented as an invited talk at the International Research Symposium on Nonstandard Analysis and Its Applications, ICMS, Edinburgh 1996, http://www.math.uha.fr/sari/papers/icms1996.pdf.Suche in Google Scholar
[23] van den Berg, I. P.: Nonstandard Asymptotic Analysis. Lecture Notes in Math. 1249, Springer-Verlag, Berlin, 1987.10.1007/BFb0077577Suche in Google Scholar
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Artikel in diesem Heft
- Regular Papers
- Semidistributivity and Whitman Property in implication zroupoids
- Composition of binary quadratic forms over number fields
- On Z-Symmetric Rings
- On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences
- Coefficient estimates for Libera type bi-close-to-convex functions
- Oscillation of nonlinear third-order differential equations with several sublinear neutral terms
- On rapidly oscillating solutions of a nonlinear elliptic equation
- Multiplicity of solutions for a class of fourth-order elliptic equations of p(x)-Kirchhoff type
- Existence of traveling wave solutions in nonlocal delayed higher-dimensional lattice systems with quasi-monotone nonlinearities
- On absolute double summability methods with high indices
- On the continuity of lattice isomorphisms on C(X, I)
- New fixed point theorems for countably condensing maps with an application to functional integral inclusions
- Common fixed point results under w-distance with applications to nonlinear integral equations and nonlinear fractional Differential Equations
- The form of locally defined operators in waterman spaces
- Conformal vector fields on almost co-Kähler manifolds
- A certain η-parallelism on real hypersurfaces in a nonflat complex space form
- On log-bimodal alpha-power distributions with application to nickel contents and erosion data
- Univariate and bivariate extensions of the generalized exponential distributions
- Pellian equations of special type
