Singularly perturbed quadratically nonlinear Dirichlet problems
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- by Albert J. DeSanti PDF
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Abstract:
The Dirichlet problem for singularly perturbed elliptic equations of the form $\varepsilon \Delta u = A({\mathbf {x}},u)\nabla u \cdot \nabla u + {\mathbf {B}}({\mathbf {x}},u) \cdot \nabla u + C({\mathbf {x}},u)$ in $\Omega \in {E^n}$ is studied. Under explicit and easily checked conditions, solutions are shown to exist for $\varepsilon$ sufficiently small and to exhibit specified asymptotic behavior as $\varepsilon \to 0$. The results are obtained using a method based on the theory of partial differential inequalities.References
- Herbert Amann, Nonlinear elliptic equations with nonlinear boundary conditions, New developments in differential equations (Proc. 2nd Scheveningen Conf., Scheveningen, 1975) North-Holland Math. Studies, Vol. 21, North-Holland, Amsterdam, 1976, pp. 43–63. MR 0509487
- W. F. Ames, Nonlinear partial differential equations in engineering, Academic Press, New York-London, 1965. MR 0210342
- Jonathan Bell, Chris Cosner, and Willy Bertiger, Solutions for a flux-dependent diffusion model, SIAM J. Math. Anal. 13 (1982), no. 5, 758–769. MR 668319, DOI 10.1137/0513052
- R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1989. Partial differential equations; Reprint of the 1962 original; A Wiley-Interscience Publication. MR 1013360 W. J. Cunningham, Electrical problems modeled by nonlinear partial differential equations, Nonlinear Partial Differential Equations, A Symposium On Methods Of Solution (W. F. Ames, ed.), Academic Press, New York, 1967. A. J. DeSanti, Boundary and interior layer behavior of solutions of some semilinear elliptic boundary value problems, J. Math. Pures Appl. (to appear).
- F. W. Dorr, S. V. Parter, and L. F. Shampine, Applications of the maximum principle to singular perturbation problems, SIAM Rev. 15 (1973), 43–88. MR 320456, DOI 10.1137/1015002
- Paul C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal. 52 (1973), 205–232. MR 374665, DOI 10.1007/BF00247733
- P. Faĭf and U. Grinli, Interior transition layers for elliptic boundary value problems with a small parameter, Uspehi Mat. Nauk 29 (1974), no. 4 (178), 103–130 (Russian). Translated from the English by L. P. Volevič; Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ, II. MR 0481510 P. Habets and M. Laloy, Etude de problemes aux limites par la methode des sur-et-sous solutions, Lecture Notes, Catholic University of Louvain, Louvain, Belgium, 1974.
- F. A. Howes, The asymptotic solution of a class of singularly perturbed nonlinear boundary value problems via differential inequalities, SIAM J. Math. Anal. 9 (1978), no. 2, 215–249. MR 477345, DOI 10.1137/0509017
- F. A. Howes, Boundary-interior layer interactions in nonlinear singular perturbation theory, Mem. Amer. Math. Soc. 15 (1978), no. 203, iv+108. MR 499407, DOI 10.1090/memo/0203
- F. A. Howes, Singularly perturbed semilinear elliptic boundary value problems, Comm. Partial Differential Equations 4 (1979), no. 1, 1–39. MR 514718, DOI 10.1080/03605307908820090
- Fred A. Howes, Some old and new results on singularly perturbed boundary value problems, Singular perturbations and asymptotics (Proc. Adv. Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1980) Publ. Math. Res. Center Univ. Wisconsin, vol. 45, Academic Press, New York-London, 1980, pp. 41–85. MR 606035
- F. A. Howes, Some singularly perturbed nonlinear boundary value problems of elliptic type, Nonlinear partial differential equations in engineering and applied science (Proc. Conf., Univ. Rhode Island, Kingston, R.I., 1979) Lecture Notes in Pure and Appl. Math., vol. 54, Dekker, New York, 1980, pp. 151–166. MR 577090
- F. A. Howes, Perturbed elliptic problems with essential nonlinearities, Comm. Partial Differential Equations 8 (1983), no. 8, 847–874. MR 699312, DOI 10.1080/03605308308820288
- Walter G. Kelley, The Dirichlet problem for singularly perturbed quasilinear elliptic equations, J. Differential Equations 40 (1981), no. 1, 37–52. MR 614217, DOI 10.1016/0022-0396(81)90009-7
- René P. Sperb, Maximum principles and their applications, Mathematics in Science and Engineering, vol. 157, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 615561
Additional Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 298 (1986), 733-746
- MSC: Primary 35B25; Secondary 35J65
- DOI: https://doi.org/10.1090/S0002-9947-1986-0860390-X
- MathSciNet review: 860390