Regular solutions of elliptic boundary-value problems with discontinuous nonlinearities
HTML articles powered by AMS MathViewer
- by
M. G. Lepchinskiĭ and V. N. Pavlenko
Translated by: I. V. Denisova - St. Petersburg Math. J. 17 (2006), 465-475
- DOI: https://doi.org/10.1090/S1061-0022-06-00915-0
- Published electronically: March 9, 2006
- PDF | Request permission
Abstract:
The existence of stable solutions to elliptic boundary-value problems is studied; stability is understood with respect to perturbations of nonlinearities. By the variational method, it is shown that stable solutions of such problems do exist provided a certain integral measure of closeness for (possibly, discontinuous) nonlinearities is employed. It is shown that problems with discontinuous nonlinearities can serve as idealization of problems with continuous nonlinearities but having narrow regions of ill-controlled variations with respect to the phase variable.References
- M. A. Krasnosel′skiĭ and A. V. Pokrovskiĭ, Regular solutions of elliptic equations with discontinuous nonlinearities, Proceedings of the All-Union Conference on Partial Differential Equations, Dedicated to the 75th Anniversary of I. G. Petrovskiĭ, Moskov. Gos. Univ., Moscow, 1978, pp. 346–347. (Russian)
- M. A. Krasnosel′skiĭ and A. V. Pokrovskiĭ, Regular solutions of equations with discontinuous nonlinearities, Dokl. Akad. Nauk SSSR 226 (1976), no. 3, 506–509 (Russian). MR 0637075
- M. A. Krasnosel′skiĭ and A. V. Pokrovskiĭ, Equations with discontinuous nonlinearities, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1056–1059 (Russian). MR 553925
- V. N. Pavlenko and R. S. Iskakov, Continuous approximations of discontinuous nonlinearities of elliptic-type semilinear equations, Ukraïn. Mat. Zh. 51 (1999), no. 2, 224–233 (Russian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 51 (1999), no. 2, 249–260. MR 1718565, DOI 10.1007/BF02513477
- M. A. Krasnosel′skiĭ and A. V. Pokrovskiĭ, Sistemy s gisterezisom, “Nauka”, Moscow, 1983 (Russian). MR 742931
- V. N. Pavlenko and V. V. Vinokur, Resonance boundary value problems for elliptic-type equations with discontinuous nonlinearities, Izv. Vyssh. Uchebn. Zaved. Mat. 5 (2001), 43–58 (Russian); English transl., Russian Math. (Iz. VUZ) 45 (2001), no. 5, 40–55. MR 1860657
- V. N. Pavlenko, Existence theorems for elliptic variational inequalities with quasipotential operators, Differentsial′nye Uravneniya 24 (1988), no. 8, 1397–1402, 1470 (Russian); English transl., Differential Equations 24 (1988), no. 8, 913–916 (1989). MR 964735
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 125307, DOI 10.1002/cpa.3160120405
Bibliographic Information
- M. G. Lepchinskiĭ
- Affiliation: Numerical Mathematics Department, Chelyabinsk State University, Brat’ev Kashirinykh Street 129, Chelyabinsk 454021, Russia
- Email: myth@csu.ru
- V. N. Pavlenko
- Affiliation: Numerical Mathematics Department, Chelyabinsk State University, Brat’ev Kashirinykh Street 129, Chelyabinsk 454021, Russia
- Received by editor(s): May 26, 2005
- Published electronically: March 9, 2006
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 17 (2006), 465-475
- MSC (2000): Primary 35J65, 35J50
- DOI: https://doi.org/10.1090/S1061-0022-06-00915-0
- MathSciNet review: 2167847