# American Institute of Mathematical Sciences

May  2013, 12(3): 1201-1220. doi: 10.3934/cpaa.2013.12.1201

## Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data

 1 Université Victor Ségalen - Bordeaux 2, 146 rue Léo Saignat, BP 26, 33076 Bordeaux 2 Centre of Mathematics for Applications, University of Oslo, P.O. Box 1053, Blindern, NO-0316 Oslo, Norway 3 Ecole Centrale de nantes, Laboratoire de Mathématiques Jean Leray, UMR CNRS 6629, 1, rue de la Noé, 44321 Nantes

Received  September 2011 Revised  August 2012 Published  September 2012

We prove existence and regularity results for distributional solutions of nonlinear elliptic and parabolic equations with general anisotropic diffusivities with variable exponents. The data are assumed to be merely integrable.
Citation: Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1201-1220. doi: 10.3934/cpaa.2013.12.1201
##### References:
 [1] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), 121-140. [2] M. Bendahmane, M. Langlais and M. Saad, On some anisotropic reaction-diffusion systems with $L^1$-data modeling the propagation of an epidemic disease, Nonlinear Anal., 54 (2003), 617-636. [3] M. Bendahmane and M. Saad, Entropy solutions for a nonlinear parabolic equation with variable exponents and L1 data,, Preprint., (). [4] M. Bendahmane and P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and $L^1$-data, Nonlinear Analysis TMA, 70 (2009), 567-583. doi: http://dx.doi.org/10.1016/j.na.2007.12.027. [5] M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1 data, Journal of Differential Equations, 249 (2010), 1483-1515. doi: DOI: 10.1016/j.jde.2010.05.011. [6] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. [7] L. Boccardo, T. Gallouët and P. Marcellini, Anisotropic equations in $L^1$, Differential Integral Equations, 9 (1996), 209-212. [8] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. [9] L. Diening, P. Hästö, T. Harjulehto and M. R.užička, "Lebesque and Sobolev Spaces with Variable Exponents," Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Berlin, 2011. [10] X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760. [11] X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(U)$ and $W^{m, p(x)}(U)$, J. Math. Anal. Appl., 263 (2001), 424-446. [12] T. Harjulehto, P. Hästö, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal., 25 (2006), 205-222. [13] O. Kovácik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1, p(x)}$, Czech. Math. J., 41 (1991), 592-618. [14] F. Li and H. Zhao, Anisotropic parabolic equations with measure data, J. Partial Differential Equations, 14 (2001), 21-30. [15] P. Marcelli, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284. [16] M. R.užička, "Electrorheological Fluids: Modeling and Mathematical Theory," Lecture Notes in Mathematics, Springer, Berlin, 2000. [17] M. Sanchon and M. Urbano, Entropy solutions for the $p(x)$-Laplace equation, Trans. American Math. Soc., 361 (2009), 6387-6405. [18] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mathematica Pura Applicata, (1987), 65-96. [19] M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi, Ricerche. Mat., 18 (1969), 3-24. [20] V. V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 67-81.

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##### References:
 [1] E. Acerbi and G. Mingione, Regularity results for a class of functionals with non-standard growth, Arch. Rational Mech. Anal., 156 (2001), 121-140. [2] M. Bendahmane, M. Langlais and M. Saad, On some anisotropic reaction-diffusion systems with $L^1$-data modeling the propagation of an epidemic disease, Nonlinear Anal., 54 (2003), 617-636. [3] M. Bendahmane and M. Saad, Entropy solutions for a nonlinear parabolic equation with variable exponents and L1 data,, Preprint., (). [4] M. Bendahmane and P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and $L^1$-data, Nonlinear Analysis TMA, 70 (2009), 567-583. doi: http://dx.doi.org/10.1016/j.na.2007.12.027. [5] M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and L1 data, Journal of Differential Equations, 249 (2010), 1483-1515. doi: DOI: 10.1016/j.jde.2010.05.011. [6] L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal., 87 (1989), 149-169. [7] L. Boccardo, T. Gallouët and P. Marcellini, Anisotropic equations in $L^1$, Differential Integral Equations, 9 (1996), 209-212. [8] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406. [9] L. Diening, P. Hästö, T. Harjulehto and M. R.užička, "Lebesque and Sobolev Spaces with Variable Exponents," Lecture Notes in Mathematics, Vol. 2017, Springer-Verlag, Berlin, 2011. [10] X. L. Fan, J. Shen and D. Zhao, Sobolev embedding theorems for spaces $W^{k,p(x)}(\Omega)$, J. Math. Anal. Appl., 262 (2001), 749-760. [11] X. L. Fan and D. Zhao, On the spaces $L^{p(x)}(U)$ and $W^{m, p(x)}(U)$, J. Math. Anal. Appl., 263 (2001), 424-446. [12] T. Harjulehto, P. Hästö, M. Koskenoja and S. Varonen, The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values, Potential Anal., 25 (2006), 205-222. [13] O. Kovácik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1, p(x)}$, Czech. Math. J., 41 (1991), 592-618. [14] F. Li and H. Zhao, Anisotropic parabolic equations with measure data, J. Partial Differential Equations, 14 (2001), 21-30. [15] P. Marcelli, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions, Arch. Ration. Mech. Anal., 105 (1989), 267-284. [16] M. R.užička, "Electrorheological Fluids: Modeling and Mathematical Theory," Lecture Notes in Mathematics, Springer, Berlin, 2000. [17] M. Sanchon and M. Urbano, Entropy solutions for the $p(x)$-Laplace equation, Trans. American Math. Soc., 361 (2009), 6387-6405. [18] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mathematica Pura Applicata, (1987), 65-96. [19] M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi, Ricerche. Mat., 18 (1969), 3-24. [20] V. V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (2004), 67-81.
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