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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 |
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. |
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), 121-140. |
[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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