Weak solutions for generalized p-Laplacian systems via Young measures

Elhoussine Azroul; Farah Balaadich

doi:10.1515/mjpaa-2018-0008

.blurhash-client-img { display: none !important; }

Weak solutions for generalized p-Laplacian systems via Young measures

Moroccan Journal of Pure and Applied Analysis

Volume 4 (2018): Issue 2 (December 2018)

By: Elhoussine Azroul and Farah Balaadich

Open Access

|May 2019

Abstract

We prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.

References

1. A. Abassi, A. El Hachimi, A. Jamea, Entropy solutions to nonlinear Neumann problems with L¹-data, Int. J. Math. Statist. 2 (2008) 4-17.
Search in Google Scholar Back to article
2. J. M. Ball, A version of the fundamental theorem for Young measures. In: PDEs and Continuum Models of Phase Transitions (Nice, 1988). Lecture Notes in Phys, 344 (1989) 207-215.10.1007/BFb0024945
Search in Google Scholar Back to article
3. D. Breit, A. Cianchi, L. Diening, T. Kuusi, S. Schwarzaker, The p-Laplace system with right-hand side in divergence form: inner and up to the boundary pointwise estimates, Nonlinear Analysis: Theory, Methods & Applications, 115(2017) 200–212.10.1016/j.na.2016.06.011
Search in Google Scholar Back to article
4. E. DiBenedetto, J. Manfredi, On the Higher Integrability of the Gradient of Weak Solutions of Certain Degenerate Elliptic Systems, American Journal of Mathematics, 115 (5) (1993) 1107-1134.10.2307/2375066
Search in Google Scholar Back to article
5. G. Dolzmann, N. Hungerbühler, S. Müller, Nonlinear elliptic systems with measure valued right hand side. Math. Z, 226 (1997) 545-574.10.1007/PL00004354
Search in Google Scholar Back to article
6. F. Duzaar, G. Mingione, Local Lipschitz regularity for degenerate elliptic systems, Ann. I. H. Poincaré AN 27 (2010) 1361-1396.10.1016/j.anihpc.2010.07.002
Search in Google Scholar Back to article
7. LC. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations. Am. Math. Soc., New York 1990.10.1090/cbms/074
Search in Google Scholar Back to article
8. N. Hungerbühler, A refinement of Ball’s theorem on Young measures, N.Y. J. Math, 3 (1997) 48-53.
Search in Google Scholar Back to article
9. T. Iwaniec, Projections onto gradients fields and L^p-estimates for degenrated elliptic operators, Studia Mathematica, 75 (3) (1983) 293-312.10.4064/sm-75-3-293-312
Search in Google Scholar Back to article
10. J. Kinuunen, S. Zhou, A local estimates for nonlinear equations with discontinuous coefficients, Commun. Partial Differential Equations 24 (11-12) (1999) 2043-2068.10.1080/03605309908821494
Search in Google Scholar Back to article
11. G M. Lieberman, The natural generalizationj of the natural conditions of ladyzhenskaya and uraltseva for elliptic equations, Commun. In. Partial Differential Equations. 16(2&3) (1991) 311-361.10.1080/03605309108820761
Search in Google Scholar Back to article
12. M. Struwe, Variational Methods, Springer Verlag Berlin, Heidelberg, New York, second edition 1996.
Search in Google Scholar Back to article
13. E. Zeidler, Nonlinear functional analysis and its application, volume I. Springer, 1986.10.1007/978-1-4612-4838-5
Search in Google Scholar Back to article