|
Regular solutions of elliptic boundary-value problems with discontinuous nonlinearities
Author(s):
M.
G.
Lepchinskii;
V.
N.
Pavlenko
Translated by:
I. V. Denisova
Original publication:
Algebra i Analiz,
tom 17
(2005),
vypusk 3.
Journal:
St. Petersburg Math. J.
17
(2006),
465-475.
MSC (2000):
Primary 35J65, 35J50
Posted:
March 9, 2006
Retrieve article in:
PDF DVI PostScript
Abstract |
References |
Similar articles |
Additional information
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:
-
- 1.
- M. A. Krasnosel'skii and A. V. Pokrovskii, 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. Petrovskii, Moskov. Gos. Univ., Moscow, 1978, pp. 346-347. (Russian)
- 2.
- -, Regular solutions of equations with discontinuous nonlinearities, Dokl. Akad. Nauk SSSR 226 (1976), no. 3, 506-509; English transl., Soviet Math. Dokl. 17 (1976), no. 1, 128-132. MR 0637075 (58:30559)
- 3.
- -, Equations with discontinuous nonlinearities, Dokl. Akad. Nauk SSSR 248 (1979), no. 5, 1056-1059; English transl., Soviet Math. Dokl. 20 (1979), no. 5, 1117-1120 (1980). MR 0553925 (81c:45010)
- 4.
- V. N. Pavlenko and R. S. Iskakov, Continuous approximations of discontinuous nonlinearities of elliptic-type semilinear equations, Ukrain. Mat. Zh. 51 (1999), no. 2, 224-233; English transl., Ukrainian Math. J. 51 (1999), no. 2, 249-260. MR 1718565 (2000j:35057)
- 5.
- M. A. Krasnosel'skii and A. V. Pokrovskii, Systems with hysteresis, ``Nauka'', Moscow, 1983. (Russian) MR 0742931 (86e:93005)
- 6.
- V. N. Pavlenko and V. V. Vinokur, Resonance boundary value problems for elliptic-type equations with discontinuous nonlinearities, Izv. Vyssh. Uchebn. Zaved. Mat. 2001, no. 5, 43-58; English transl., Russian Math. (Iz. VUZ) 45 (2001), no. 5, 40-55. MR 1860657 (2002h:35100)
- 7.
- V. N. Pavlenko, Existence theorems for elliptic variational inequalities with quasipotential operators, Differentsial'nye Uravneniya 24 (1988), no. 8, 1397-1402; English transl., Differential Equations 24 (1988), no. 8, 913-916 (1989). MR 0964735 (90a:35101)
- 8.
- 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 0125307 (23:A2610)
Similar Articles:
Retrieve articles in St. Petersburg Mathematical Journal
with MSC
(2000):
35J65, 35J50
Retrieve articles in all Journals with MSC
(2000):
35J65, 35J50
Additional Information:
M.
G.
Lepchinskii
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
DOI:
10.1090/S1061-0022-06-00915-0
PII:
S 1061-0022(06)00915-0
Keywords:
Elliptic boundary-value problems,
discontinuous nonlinearities,
strong solutions,
$\mathrm{nl}$-stable solutions
Received by editor(s):
26/MAY/2005
Posted:
March 9, 2006
Copyright of article:
Copyright
2006,
American Mathematical Society
|
Comments: webmaster@ams.org