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Renormalized solutions to a reaction-diffusion system applied to image denoising
1. | College of Mathematics and Computational Science, Shenzhen University, 518060 Shenzhen, China |
2. | Department of Mathematics, Harbin Institute of Technology, 150001 Harbin, China |
3. | School of Mathematics, Jilin University, Changchun 130012 |
References:
[1] |
R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.
doi: 10.1016/j.camwa.2008.01.017. |
[2] |
F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Renormalized solutions for degenerate elliptic-parabolic problems with nonlinear dynamical boundary condtions and $L^1$-data, Journal of differential equations, 244 (2008), 2764-2803.
doi: 10.1016/j.jde.2008.02.022. |
[3] |
F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Advances in mathematical sciences and applications, 7 (1997), 183-213. |
[4] |
F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Existence and uniqueness for a degenerate parabolic equation with $L^1$ data, Trans. Amer. Math. Soc, 351 (1999), 285-306.
doi: 10.1090/S0002-9947-99-01981-9. |
[5] |
F. Andreu, J. M. Mazón and J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces and Free Boundaries, 8 (2006), 447-479.
doi: 10.4171/IFB/151. |
[6] |
M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^1$-data, J. Differential Equations, 249 (2010), 1483-1515.
doi: 10.1016/j.jde.2010.05.011. |
[7] |
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241-273. |
[8] |
D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with $L^1$ data, existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.
doi: 10.1017/S0308210500026986. |
[9] |
D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.
doi: 10.1006/jdeq.2000.4013. |
[10] |
L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1998), 183-196.
doi: 10.1007/BF01766148. |
[11] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[12] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
[13] |
R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.
doi: 10.2307/1971423. |
[14] |
J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, Nonlinear Differential Equations Appl., 14 (2007), 181-205.
doi: 10.1007/s00030-007-5018-z. |
[15] |
R. C. Gonzalez and R. E. Woods, Digital Image Processing, $2^{nd}$ edition, Pearson Prentice Hall, 2002. |
[16] |
Y. Gousseau and J. M. Morel, Are natural images of bounded variation?, SIAM Journal on Mathematical Analysis, 33 (2001), 634-648.
doi: 10.1137/S0036141000371150. |
[17] |
Z. C. Guo, Q. Liu, J. B. Sun and B. Y. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Analysis: Real World Applications, 12 (2011), 2904-2918.
doi: 10.1016/j.nonrwa.2011.04.015. |
[18] |
Z. C. Guo, J. B. Sun, D. Z. Zhang and B. Y. Wu, Adaptive Perona-Malik model based on the variable exponent for image denoising, IEEE Transactions on Image Processing, 21 (2012), 958-967.
doi: 10.1109/TIP.2011.2169272. |
[19] |
Z. C. Guo, J. X. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Mathematical and Computer Modelling, 53 (2011), 1336-1350.
doi: 10.1016/j.mcm.2010.12.031. |
[20] |
R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect A, 89 (1981), 217-237.
doi: 10.1017/S0308210500020242. |
[21] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1968. |
[22] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[23] |
I. Nyanquini and S. Ouaro, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afrika Matematika, 23 (2012), 205-228.
doi: 10.1007/s13370-011-0030-1. |
[24] |
A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations, Ann. Mat. Pura ed Applicata, 177 (1999), 143-172.
doi: 10.1007/BF02505907. |
[25] |
A. Porretta, Regularity for entropy solutions of a class of parabolic equations with nonregular initial datum, Dynam. Systems Appl., 7 (1998), 53-71. |
[26] |
H. Redwane, Existence of a solution for a class of nonlinear parabolic systems, Electron. J. Qual. Theory Differ. Equ, 24 (2007), 1-18. |
[27] |
S. Segura de Lenón and J. Toledo, Regularity for entropy solutions of parabolic $p$-Laplacian type equations, Publ. Mat., 43 (1999), 665-683.
doi: 10.5565/PUBLMAT_43299_08. |
[28] |
Z. Q. Wu, J. Y. Yin, H. L. Li and J. N. Zhao, Nonlinear Diffusion Equations, World Scientific Publishing Company, 2001.
doi: 10.1142/9789812799791. |
[29] |
C. Zhang and S. L. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$-data, J. Differential Equations, 248 (2010), 1376-1400.
doi: 10.1016/j.jde.2009.11.024. |
show all references
References:
[1] |
R. Aboulaich, D. Meskine and A. Souissi, New diffusion models in image processing, Comput. Math. Appl., 56 (2008), 874-882.
doi: 10.1016/j.camwa.2008.01.017. |
[2] |
F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, Renormalized solutions for degenerate elliptic-parabolic problems with nonlinear dynamical boundary condtions and $L^1$-data, Journal of differential equations, 244 (2008), 2764-2803.
doi: 10.1016/j.jde.2008.02.022. |
[3] |
F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Quasi-linear elliptic and parabolic equations in $L^1$ with nonlinear boundary conditions, Advances in mathematical sciences and applications, 7 (1997), 183-213. |
[4] |
F. Andreu, J. M. Mazón, S. Segura de León and J. Toledo, Existence and uniqueness for a degenerate parabolic equation with $L^1$ data, Trans. Amer. Math. Soc, 351 (1999), 285-306.
doi: 10.1090/S0002-9947-99-01981-9. |
[5] |
F. Andreu, J. M. Mazón and J. Toledo, A degenerate elliptic-parabolic problem with nonlinear dynamical boundary conditions, Interfaces and Free Boundaries, 8 (2006), 447-479.
doi: 10.4171/IFB/151. |
[6] |
M. Bendahmane, P. Wittbold and A. Zimmermann, Renormalized solutions for a nonlinear parabolic equation with variable exponents and $L^1$-data, J. Differential Equations, 249 (2010), 1483-1515.
doi: 10.1016/j.jde.2010.05.011. |
[7] |
P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22 (1995), 241-273. |
[8] |
D. Blanchard and F. Murat, Renormalized solutions of nonlinear parabolic problems with $L^1$ data, existence and uniqueness, Proc. Roy. Soc. Edinburgh Sect. A, 127 (1997), 1137-1152.
doi: 10.1017/S0308210500026986. |
[9] |
D. Blanchard, F. Murat and H. Redwane, Existence and uniqueness of a renormalized solution for a fairly general class of nonlinear parabolic problems, J. Differential Equations, 177 (2001), 331-374.
doi: 10.1006/jdeq.2000.4013. |
[10] |
L. Boccardo, F. Murat and J. P. Puel, Existence of bounded solutions for nonlinear elliptic unilateral problems, Annali di Matematica Pura ed Applicata, 152 (1998), 183-196.
doi: 10.1007/BF01766148. |
[11] |
Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.
doi: 10.1137/050624522. |
[12] |
L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-18363-8. |
[13] |
R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. Math., 130 (1989), 321-366.
doi: 10.2307/1971423. |
[14] |
J. Droniou and A. Prignet, Equivalence between entropy and renormalized solutions for parabolic equations with smooth measure data, Nonlinear Differential Equations Appl., 14 (2007), 181-205.
doi: 10.1007/s00030-007-5018-z. |
[15] |
R. C. Gonzalez and R. E. Woods, Digital Image Processing, $2^{nd}$ edition, Pearson Prentice Hall, 2002. |
[16] |
Y. Gousseau and J. M. Morel, Are natural images of bounded variation?, SIAM Journal on Mathematical Analysis, 33 (2001), 634-648.
doi: 10.1137/S0036141000371150. |
[17] |
Z. C. Guo, Q. Liu, J. B. Sun and B. Y. Wu, Reaction-diffusion systems with $p(x)$-growth for image denoising, Nonlinear Analysis: Real World Applications, 12 (2011), 2904-2918.
doi: 10.1016/j.nonrwa.2011.04.015. |
[18] |
Z. C. Guo, J. B. Sun, D. Z. Zhang and B. Y. Wu, Adaptive Perona-Malik model based on the variable exponent for image denoising, IEEE Transactions on Image Processing, 21 (2012), 958-967.
doi: 10.1109/TIP.2011.2169272. |
[19] |
Z. C. Guo, J. X. Yin and Q. Liu, On a reaction-diffusion system applied to image decomposition and restoration, Mathematical and Computer Modelling, 53 (2011), 1336-1350.
doi: 10.1016/j.mcm.2010.12.031. |
[20] |
R. Landes, On the existence of weak solutions for quasilinear parabolic initial-boundary value problems, Proc. Roy. Soc. Edinburgh Sect A, 89 (1981), 217-237.
doi: 10.1017/S0308210500020242. |
[21] |
O. A. Ladyženskaja, V. A. Solonnikov and N. N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society: Providence, RI, 1968. |
[22] |
J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Gauthier-Villars, Paris, 1969. |
[23] |
I. Nyanquini and S. Ouaro, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afrika Matematika, 23 (2012), 205-228.
doi: 10.1007/s13370-011-0030-1. |
[24] |
A. Porretta, Existence results for nonlinear parabolic equations via strong convergence of trauncations, Ann. Mat. Pura ed Applicata, 177 (1999), 143-172.
doi: 10.1007/BF02505907. |
[25] |
A. Porretta, Regularity for entropy solutions of a class of parabolic equations with nonregular initial datum, Dynam. Systems Appl., 7 (1998), 53-71. |
[26] |
H. Redwane, Existence of a solution for a class of nonlinear parabolic systems, Electron. J. Qual. Theory Differ. Equ, 24 (2007), 1-18. |
[27] |
S. Segura de Lenón and J. Toledo, Regularity for entropy solutions of parabolic $p$-Laplacian type equations, Publ. Mat., 43 (1999), 665-683.
doi: 10.5565/PUBLMAT_43299_08. |
[28] |
Z. Q. Wu, J. Y. Yin, H. L. Li and J. N. Zhao, Nonlinear Diffusion Equations, World Scientific Publishing Company, 2001.
doi: 10.1142/9789812799791. |
[29] |
C. Zhang and S. L. Zhou, Renormalized and entropy solutions for nonlinear parabolic equations with variable exponents and $L^1$-data, J. Differential Equations, 248 (2010), 1376-1400.
doi: 10.1016/j.jde.2009.11.024. |
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