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Quasilinear Degenerated Elliptic Systems with Weighted in Divergence Form with Weak Monotonicity with General Data
Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco
Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco
Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco
- 1 Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco
- 2 Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco
- 3 Laboratory LAMA, Department of Mathematics, Faculty of Sciences Dhar El Mahraz, Sidi Mohammed Ben Abdellah University, Atlas Fez, Morocco
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Abstract
We consider, for a bounded open domain Ω in R n and a function u : Ω → R m , the quasilinear elliptic system: (1). We generalize the system ( QES ) ( f , g ) in considering a right hand side depending on the jacobian matrix Du . Here, the star in ( QES ) ( f , g ) indicates that f may depend on Du . In the right hand side, v belongs to the dual space W -1, P ’ (Ω, ω * , R m ), , f and g satisfy some standard continuity and growth conditions. We prove existence of a regularity, growth and coercivity conditions for σ , but with only very mild monotonicity assumptions.
KeywordsQuasilinear EllipticSobolev Spaces with WeightYoung MeasureGalerkin Scheme
- Ball, J. (1989) A Version of the Fundamental Theorem for Young Measures. In Partial Differential Equations and Continuum Models of Phase Transitions. Proceedings of an NSF-CNRS Joint Seminar, Nice, France, 207-215.
- Dolzmann, G., Hungerbuhler, N. and Muller, S. (1997) Nonlinear Elliptic Systems with Measure-Valued Right Hand Side. Mathematische Zeitschrift, 226, 545-574. https://doi.org/10.1007/PL00004354
- Minty, G.J. (1962) Monotone (Nonlinear) Operators in Hilbert Space. Duke Mathematical Journal, 29, 341-346. https://doi.org/10.1215/S0012-7094-62-02933-2
- Brezis, H. (1973) Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, No. 5. Notas de Matematica (50). North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co. Inc., New York.
- Edwin, H. and Stromberg, K. (1969) Real and Abstract Analysis. Springer-Verlag, Berlin.
- Fonseca, I., Muller, S. and Pedregal, P. (1998) Analysis of Concentration and Oscillation Effects Generated by Gradients. SIAM Journal on Mathematical Analysis, 29, 736-756. https://doi.org/10.1137/S0036141096306534
- Lions, J.L. (1969) Quelques Methodes de Resolution des Problemes aux Limites non Lineaires. Dunod, Gauthier-Villars, Paris.
- Kristznsen, J. (1999) Lower Semicontinuity in Space of Weakly Differentiable Functions. Mathematische Annalen, 313, 653-710. https://doi.org/10.1007/s002080050277
- Visik, M.I. (1963) Quasi-Linear Strongly Elliptic Systems of Differential Equations of Divergence Form. Trudy Moskov. Mat. Obsc, 12, 125-184. (In Russian)
- Hungerbühler, N. (1999) Quasilinear Elliptic Systems in Divergence form with Weak Monotonicity. New York Journal of Mathematics, 5, 83-90.
- Akdim, Y. and Azroul, E. (2002) Pseudo-Monotonicity and Degenerated Elliptic Operator of Second Order. Southwest Texas State University, San Marcos.
- Benilan, P., Brezis, H. and Crandall, M.G (1975) A Semilinear Equation in L1. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 2, 523-555.
- Fabrie, P. and Gallouet. T (2000) Modelling Wells in Porous Media Flows. Mathematical Models and Methods in Applied Sciences, 10, 673-709. https://doi.org/10.1142/S0218202500000367
- Kinderlehrer, D. and Pedregal, P. (1994) Gradient Young Measures Generated by Sequences in Sobolev Spaces. The Journal of Geometric Analysis, 4, Article No. 59. https://doi.org/10.1007/BF02921593