Existence of weak solutions for some quasilinear degenerated elliptic systems in weighted Sobolev spaces
Keywords:quasilinear elliptic system, Young measure, Galerkin schema
We consider, for a bounded open domain Ω in Rn; (n ≥ 1) and a function u : Ω → ℝm; (m ≥ 1) the quasilinear elliptic system:
Which is a Dirichlet problem. Here, v belongs to the dual space , f and g satisfy some stan- dard continuity and growth conditions. we will show the existence of a weak solution of this problem in the four following cases: σ is mono- tonic, σ is strictly monotonic, σ is quasi montone and σ derives from a convex potential.
J. Ball, “A version of the fundamental theorem for Young measures,” in PDEs and Continuum Models of Phase Transitions: Proceedings of an NSF-CNRS Joint Seminar Held in Nice, France, January 18–22, 1988, M. Rascle, D. Serre and M. Slemrod, Eds. Berlin: Springer, 1989, pp. 2017-215.
D. Kinderlehrer, P. Pedregal, “Gradient Young measures generated by sequences in Sobolev spaces”, Journal of Geometric Analysis, vol. 4, no. 1, pp. 59-90, 1994. https://doi.org/10.1007/bf02921593
F. E. Browder, “Existence theorems for nonlinear partial differential equations”, Global Analysis, pp. 1–60, 1970. https://doi.org/10.1090/pspum/016/0269962
G. Dolzmann, N. Hungerbühler and S. Muller, “Nonlinear elliptic systems with measure-valued right hand side”, Mathematische Zeitschrift, vol. 226, pp. 545-574, 1997. https://doi.org/10.1007/pl00004354
G. J. Minty, “Monotone (nonlinear) operators in Hilbert space”, Duke Mathematical Journal, vol. 29, pp. 341-346, 1962. https://doi.org/10.1215/s0012-7094-62-02933-2
H. Brezis, Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. Amsterdam: Elsevier, 1973.
E. Hewitt and K.R. Stromberg, Real and abstract analysis. Berlin: Springer, 1969.
I. Fonseca, S. Muller and P. Pedregal, “Analysis of concentration and oscillation effects generated by gradients”, SIAM Journal on Mathematical Analysis, vol. 29, pp. 736-756, 1998. https://doi.org/10.1137/s0036141096306534
J. L. Lions, Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Paris: Dunod, Gauthier-Villars, 1969.
J. Kristensen, “Lower semicontinuity in space of weakly differentiable functions”, Mathematische Annalen, vol. 313, no. 4, pp. 653-710, 1999. https://doi.org/10.1007/s002080050277
L. Boccardo and F. Murat, “Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations”, Nonlinear Analysis: Theory, Methods & Applications, vol. 19, no. 6, pp. 581-597, 1992. https://doi.org/10.1016/0362-546x(92)90023-8
M. Candela, E. Medeiros, G. Palmieri and K. Perera, “Weak solutions of quasilinear elliptic systems via the cohomological index”, Topological Methods in Nonlinear Analysis, vol. 36, no. 1, pp. 1-18, 2010. [On line]. Available: https://bit.ly/3UGYdQa
M. I. Visik, “Quasi-linear strongly elliptic systems of differential equations of divergence form”, Trudy Moskovskogo Matematicheskogo Obshchestva, vol. 12, pp. 125-184, 1963. [On line]. Available: https://bit.ly/3Eg0Thh
N. Hungerbuhler, “Quasilinear elliptic systems in divergence form with weak monotonicity”, New York Journal of Mathematics , vol. 5, pp. 83-90, 1999. [On line]. Available: https://bit.ly/3Elrwm7
P. Drabek, A. Kufnen and V. Mustonen, “Pseudo-Monotinicity and degenerated, a singular operators”, Bulletin of the Australian Mathematical Society, vol. 58, pp. 213-221, 1998. https://doi.org/10.1017/s0004972700032184
R. Landes and V. Mustonen, “On pseudomonotone operators and nonlinear noncoercive variational problems on unbounded domains”, Mathematische Annalen, vol. 248, pp. 241-246, 1980. https://doi.org/10.1007/bf01420527
Y. Akdim and E. Azroul, “Pseudo-monotonicity and degenerated elliptic operator of second order”, Electronic Journal of Differential Equations, no. 09, pp 9–24, 2002.
Y. Akdim, E.H. Azroul and A. Benkirane, “Existence of solutions for quasilinear degenerate elliptic equations”, Electronic Journal of Differential Equations, no. 71, pp. 19, 2001.
E. Zeidler, Nonlinear functional Analysis and its application, New York: Springer, 1986.
How to Cite
Copyright (c) 2022 Rami El Houcine, Azroul Elhoussine , Barbara Abdelkrim
This work is licensed under a Creative Commons Attribution 4.0 International License.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.