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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

Mostafa Bendahmane 1, , Kenneth Hvistendahl Karlsen 2, and  Mazen Saad 3,

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.
Keywords: anisotropic diffusion, parabolic equation, $L^1$ data., weak solutions, Elliptic equation, variable exponents.
Mathematics Subject Classification: 35J60, 35K55, 35D1.
Citation: Mostafa Bendahmane, Kenneth Hvistendahl Karlsen, Mazen Saad. Nonlinear anisotropic elliptic and parabolic equations with variable exponents and $L^1$ data. Communications on Pure & 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.   Google Scholar

[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.   Google Scholar

[3]

M. Bendahmane and M. Saad, Entropy solutions for a nonlinear parabolic equation with variable exponents and L1 data,, Preprint., ().   Google Scholar

[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.  doi: http://dx.doi.org/10.1016/j.na.2007.12.027.  Google Scholar

[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.  doi: DOI: 10.1016/j.jde.2010.05.011.  Google Scholar

[6]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149.   Google Scholar

[7]

L. Boccardo, T. Gallouët and P. Marcellini, Anisotropic equations in $L^1$,, Differential Integral Equations, 9 (1996), 209.   Google Scholar

[8]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math., 66 (2006), 1383.   Google Scholar

[9]

L. Diening, P. Hästö, T. Harjulehto and M. R.užička, "Lebesque and Sobolev Spaces with Variable Exponents,", Lecture Notes in Mathematics, (2017).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

O. Kovácik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1, p(x)}$,, Czech. Math. J., 41 (1991), 592.   Google Scholar

[14]

F. Li and H. Zhao, Anisotropic parabolic equations with measure data,, J. Partial Differential Equations, 14 (2001), 21.   Google Scholar

[15]

P. Marcelli, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions,, Arch. Ration. Mech. Anal., 105 (1989), 267.   Google Scholar

[16]

M. R.užička, "Electrorheological Fluids: Modeling and Mathematical Theory,", Lecture Notes in Mathematics, (2000).   Google Scholar

[17]

M. Sanchon and M. Urbano, Entropy solutions for the $p(x)$-Laplace equation,, Trans. American Math. Soc., 361 (2009), 6387.   Google Scholar

[18]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mathematica Pura Applicata, (1987), 65.   Google Scholar

[19]

M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi,, Ricerche. Mat., 18 (1969), 3.   Google Scholar

[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.   Google Scholar

show all references

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.   Google Scholar

[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.   Google Scholar

[3]

M. Bendahmane and M. Saad, Entropy solutions for a nonlinear parabolic equation with variable exponents and L1 data,, Preprint., ().   Google Scholar

[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.  doi: http://dx.doi.org/10.1016/j.na.2007.12.027.  Google Scholar

[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.  doi: DOI: 10.1016/j.jde.2010.05.011.  Google Scholar

[6]

L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data,, J. Funct. Anal., 87 (1989), 149.   Google Scholar

[7]

L. Boccardo, T. Gallouët and P. Marcellini, Anisotropic equations in $L^1$,, Differential Integral Equations, 9 (1996), 209.   Google Scholar

[8]

Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration,, SIAM J. Appl. Math., 66 (2006), 1383.   Google Scholar

[9]

L. Diening, P. Hästö, T. Harjulehto and M. R.užička, "Lebesque and Sobolev Spaces with Variable Exponents,", Lecture Notes in Mathematics, (2017).   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[13]

O. Kovácik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{1, p(x)}$,, Czech. Math. J., 41 (1991), 592.   Google Scholar

[14]

F. Li and H. Zhao, Anisotropic parabolic equations with measure data,, J. Partial Differential Equations, 14 (2001), 21.   Google Scholar

[15]

P. Marcelli, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth conditions,, Arch. Ration. Mech. Anal., 105 (1989), 267.   Google Scholar

[16]

M. R.užička, "Electrorheological Fluids: Modeling and Mathematical Theory,", Lecture Notes in Mathematics, (2000).   Google Scholar

[17]

M. Sanchon and M. Urbano, Entropy solutions for the $p(x)$-Laplace equation,, Trans. American Math. Soc., 361 (2009), 6387.   Google Scholar

[18]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mathematica Pura Applicata, (1987), 65.   Google Scholar

[19]

M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi,, Ricerche. Mat., 18 (1969), 3.   Google Scholar

[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.   Google Scholar

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