On Some Elliptic Problems in Unbounded Domains

Abstract

The author presents a method allowing to obtain existence of a solution for some elliptic problems set in unbounded domains, and shows exponential rate of convergence of the approximate solution toward the solution.

Keywords

Elliptic / Exponential convergence / Unbounded domains

References

[1] Brezis H., Chipot M., Xie Y.. Some remarks on Liouville type theorems. Proceedings of the International Conference in Nonlinear Analysis, 2008, Hsinchu, Taiwan: World Scientific Edt 43-65

[2] Chipot M.. l Goes to Plus Infinity, 2002, Birkhäuser, Basel: Birkhäuser Advanced Texts

[3] Chipot M.. Elliptic Equations: An Introductory Course, 2009, Birkhäuser, Basel: Birkhäuser Advanced Texts

[4] Chipot M.. Asymptotic Issues for Some Partial Differential Equations, 2016, London: Imperial College Press

[5] Chipot M., Mardare S.. Asymptotic behaviour of the Stokes problem in cylinders becoming unbounded in one direction. J. Math. Pures Appl., 2008, 90(2): 133-159

[6] Chipot M., Mardare S.. The Neumann problem in cylinders becoming unbounded in one direction. J. Math. Pures Appl., 2015, 104(5): 921-941

[7] Chipot M., Rougirel A.. On the asymptotic behaviour of the solution of elliptic problems in cylindrical domains becoming unbounded. Commun. Contemp. Math., 2002, 4(1): 15-44

[8] Chipot M., Yeressian K.. Exponential rates of convergence by an iteration technique. C. R. Acad. Sci. Paris, Ser. I, 2008, 346: 21-26

[9] Chipot M., Yeressian K.. On some variational inequalities in unbounded domains. Boll. Unione Mat. Ital., 2012, 9: 243-262

[10] Chipot, M. and Zube, S., On the asymptotic behaviour of the pure Neumann problem in cylinder-like domains and its applications, Asymptotic Analysis, preprint.

[11] Dautray R., Lions J.-L.. Analyse Mathématique et Calcul Numérique Pour les Sciences et les Techniques, 1984

[12] Diaz J. I., Oleinik O. A.. Nonlinear elliptic boundary-value problems in unbounded domain and the asymptotic behavior of its solution. C. R. Acad. Sci. Paris, 1992, 315(1): 787-792

[13] Evans, L. C., Partial Differential Equations, Graduate Studies inMathematics, 19, AmericanMathematical Society, Providence, 1998.

[14] Gilbarg D., Trudinger N. S.. Elliptic Partial Differential Equations of Second Order, 2001, Berlin, Heidelberg, New York: Springer-Verlag

[15] Kinderlehrer D., Stampacchia G.. An Introduction to Variational Inequalities and Their Applications, 1980, New York: Academic Press

[16] Kondratiev V. A., Oleinik O. A.. On asymptotic behavior of solutions of some nonlinear elliptic equations in unbounded domains, Partial differential equations and related subjects. Proceedings of the conference dedicated to Louis Nirenberg, 1992 163-195

[17] Oleinik O. A.. Some Asymptotic Problems in the Theory of Partial Differential Equations, 1996