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Double phase variable exponent problems with nonlinear matrices diffusion

Year 2025, Volume: 54 Issue: 2, 445 - 456, 28.04.2025

Abderrahim Charkaoui , Jinlan Pan

https://doi.org/10.15672/hujms.1428174
Cited By: 2

Abstract

This work tackles a class of double phase elliptic problems with variable exponents and matrices diffusion. Under suitable assumptions on the data, we use critical point theory to establish both the existence and uniqueness of weak solutions to the double phase problem under consideration.

Keywords

double phase problem variable exponent weak solution matrices diffusion

References

  • [1] H. Alaa, N. E. Alaa, A. Bouchriti and A. Charkaoui, An improved nonlinear anisotropic model with $p(x)$-growth conditions applied to image restoration and enhancement, Math. Meth. Appl. Sci. 47 (9), 7546–7575, 2024.
  • [2] H. Alaa, N. E. Alaa and A. Charkaoui, Time periodic solutions for strongly nonlinear parabolic systems with $p(x)$-growth conditions, J. Elliptic Parabol. Equ. 7, 815–839, 2021.
  • [3] N. E. Alaa, A. Charkaoui, M. El Ghabi and M. El Hathout, Integral Solution for a Parabolic Equation Driven by the $p(x)$-Laplacian Operator with Nonlinear Boundary Conditions and L1 Data, Mediterr. J. Math. 20 (5), 244, 2023.
  • [4] A. Alvino, V. Ferone and G. Trombetti, On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal. 29, 437–451, 1998.
  • [5] A. Bahrouni, V. D. Rˇadulescu and D.D. Repovš, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32, 2481–2495, 2019.
  • [6] A. Bahrouni, V. D. Rˇadulescu and D. D. Repovš, A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications, Nonlinearity, 31, 1516–1534, 2018.
  • [7] P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121, 206–222, 2015.
  • [8] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. PDEs 57, 48 pp. 2018.
  • [9] S. S. Byun, J. Ok, K. Song, Hölder regularity for weak solutions to nonlocal double phase problems, J. Math. Pures Appl. 168, 110–142, 2022.
  • [10] J. X. Cen, S. J. Kim, Y. H. Kim and S. Zeng, Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent, Adv. Differential Equ. 28, 467–504, 2023.
  • [11] M. Cencelj, V. D. Radulescu and D. D. Repov˘s, Double phase problems with variable growth, Nonlinear Anal. 177, 270–287, 2018.
  • [12] A. Charkaoui, A. Ben-Loghfyry and S. Zeng, A Novel Parabolic Model Driven by Double Phase Flux Operator with Variable Exponents: Application to Image Decomposition and Denoising, Available at SSRN 4682810.
  • [13] A. Charkaoui, Periodic solutions for nonlinear evolution equations with $p(x)$-growth structure, Evol. Equ. Control Theory 13 (3), 877-892, 2024.
  • [14] A. Charkaoui and N. E. Alaa, Existence and uniqueness of renormalized periodic solution to a nonlinear parabolic problem with variable exponent and L1 data. J. Math. Anal. Appl. 506 (2), 125674, 2022.
  • [15] A. Charkaoui and N. E. Alaa, An $L^1$-theory for a nonlinear temporal periodic problem involving $p(x)$-growth structure with a strong dependence on gradients, J. Evol. Equ. 23, 73, 2023.
  • [16] A. Charkaoui, A. Ben-Loghfyry and S. Zeng, Nonlinear Parabolic Double Phase Variable Exponent Systems with Applications in Image Noise Removal, Applied Mathematical Modelling, 132, 495–530, 2024.
  • [17] A. Charkaoui, H. Fahim and N. E. Alaa, Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent, Opuscula Math. 41, 25–53, 2021.
  • [18] Y. Chen, S. Levineand and M. Rao, Variable exponent linear growth functionals in image restoration, SIAM J. Appl. Math. 66, 1383–1406, 2006.
  • [19] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215, 443–496, 2015.
  • [20] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218, 219–273, 2015.
  • [21] A. Crespo-Blanco, L. Gasinski, P. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: existence and uniqueness, J. Differerential Equ. 323, 182–228, 2022.
  • [22] C. De Filippis and G. Mingione, Regularity for double phase problems at nearly linear growth, Arch. Ration. Mecha. Anal. 247, 50 pp. 2023.
  • [23] A. El Khalil, P. Lindqvist and A. Touzani, On the stability of the first eigenvalue of the problem: $A_{p}u+\lambda g(x)|u|^{p-2}u=0$ with varying p, Rend. Mat. 24, 321–336, 2004.
  • [24] H. Fahim, A. Charkaoui and N. E. Alaa, Parabolic systems driven by general differential operators with variable exponents and strong nonlinearities with respect to the gradient. J Elliptic Parabol. Equ. 7, 199–219, 2021.
  • [25] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263, 424–446, 2001.
  • [26] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367, 204–228, 2010.
  • [27] L. Gasinski and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differ. Equ. 268, 4183–4193, 2020.
  • [28] K. Ho and P. Winkert, New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems, Calc. Var. PDEs 62, 38 pp. 2023.
  • [29] I. H. Kim, Y. H. Kim, M. W. Oh and S. Zeng, Existence and multiplicity of solutions to concave-convex-type double-phase problems with variable exponent, Nonlinear Anal. 67, 103627, 2022.
  • [30] O. Kovácik and J. Rákosník, On spaces $L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$, Czechoslovak Math. J. 41, 592–618, 1991.
  • [31] W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differ. Equ. 265, 4311–4334, 2018.
  • [32] W. Liu and G. Dai, Three ground state solutions for double phase problem, J. Math. Phys. 59, 121503, 2018.
  • [33] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Ration. Mech. Anal. 105, 267–284, 1989.
  • [34] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, qgrowth conditions, J. Differ. Equ. 90, 1–30, 1991.
  • [35] M. Mihˇailescu and V. Rˇadulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A 462, 2625–2641, 2006.
  • [36] M. Mihˇailescu and D. Repovš, On a PDE involving the $\mathcal{A}_{p(\cdot)}$-Laplace operator, Nonlinear Anal. 75, 975–981, 2012.
  • [37] N. S. Papageorgiou, V. D. Radulescu and D. D. Repov˘s, Double-phase problems and a discontinuity property of the spectrum, P. Am. Math. Soc. 147, 2899–2910, 2019.
  • [38] V. D. Radulescu and D. D. Repov˘s, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press Taylor and Francis Group, 2015.
  • [39] K. Rajagopal, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn. 13, 59–78, 2001.
  • [40] Y. G. Reshetnyak, Set of singular points of solutions of certain nonlinear elliptic equations, Sibirsk. Mat. Zh. 9, 354–368, 1968. (in Russian).
  • [41] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science and Business Media, 2000.
  • [42] S. Zeng, N. S. Papageorgiou and P. Winkert, Inverse problems for double-phase obstacle problems with variable exponents, J. Optim. Theory Appl. 196, 666–699, 2023.
  • [43] S. Zeng, V. D. Radulescu and P. Winkert, Double phase obstacle problems with variable exponent, Adv. Differential Equ. 27, 611–645, 2022.
  • [44] S. Zeng, V. D. Radulescu and P. Winkert, Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions, SIAM J. Math. Anal. 54, 1898–1926, 2022.
  • [45] Q. Zhang and V. D. Rˇadulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. 118, 159–203, 2018.
  • [46] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Ros. Akadem. Nauk. 50, 675–710, 1986.
  • [47] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phy. 3, 2 pp. 1995.
  • [48] V. V. Zhikov, On some variational problems, Russian J. Math. Phys. 5, 105 pp. 1997.

Year 2025, Volume: 54 Issue: 2, 445 - 456, 28.04.2025

Abderrahim Charkaoui , Jinlan Pan

https://doi.org/10.15672/hujms.1428174
Cited By: 2

Abstract

References

  • [1] H. Alaa, N. E. Alaa, A. Bouchriti and A. Charkaoui, An improved nonlinear anisotropic model with $p(x)$-growth conditions applied to image restoration and enhancement, Math. Meth. Appl. Sci. 47 (9), 7546–7575, 2024.
  • [2] H. Alaa, N. E. Alaa and A. Charkaoui, Time periodic solutions for strongly nonlinear parabolic systems with $p(x)$-growth conditions, J. Elliptic Parabol. Equ. 7, 815–839, 2021.
  • [3] N. E. Alaa, A. Charkaoui, M. El Ghabi and M. El Hathout, Integral Solution for a Parabolic Equation Driven by the $p(x)$-Laplacian Operator with Nonlinear Boundary Conditions and L1 Data, Mediterr. J. Math. 20 (5), 244, 2023.
  • [4] A. Alvino, V. Ferone and G. Trombetti, On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal. 29, 437–451, 1998.
  • [5] A. Bahrouni, V. D. Rˇadulescu and D.D. Repovš, Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves, Nonlinearity, 32, 2481–2495, 2019.
  • [6] A. Bahrouni, V. D. Rˇadulescu and D. D. Repovš, A weighted anisotropic variant of the Caffarelli-Kohn-Nirenberg inequality and applications, Nonlinearity, 31, 1516–1534, 2018.
  • [7] P. Baroni, M. Colombo and G. Mingione, Harnack inequalities for double phase functionals, Nonlinear Anal. 121, 206–222, 2015.
  • [8] P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calc. Var. PDEs 57, 48 pp. 2018.
  • [9] S. S. Byun, J. Ok, K. Song, Hölder regularity for weak solutions to nonlocal double phase problems, J. Math. Pures Appl. 168, 110–142, 2022.
  • [10] J. X. Cen, S. J. Kim, Y. H. Kim and S. Zeng, Multiplicity results of solutions to the double phase anisotropic variational problems involving variable exponent, Adv. Differential Equ. 28, 467–504, 2023.
  • [11] M. Cencelj, V. D. Radulescu and D. D. Repov˘s, Double phase problems with variable growth, Nonlinear Anal. 177, 270–287, 2018.
  • [12] A. Charkaoui, A. Ben-Loghfyry and S. Zeng, A Novel Parabolic Model Driven by Double Phase Flux Operator with Variable Exponents: Application to Image Decomposition and Denoising, Available at SSRN 4682810.
  • [13] A. Charkaoui, Periodic solutions for nonlinear evolution equations with $p(x)$-growth structure, Evol. Equ. Control Theory 13 (3), 877-892, 2024.
  • [14] A. Charkaoui and N. E. Alaa, Existence and uniqueness of renormalized periodic solution to a nonlinear parabolic problem with variable exponent and L1 data. J. Math. Anal. Appl. 506 (2), 125674, 2022.
  • [15] A. Charkaoui and N. E. Alaa, An $L^1$-theory for a nonlinear temporal periodic problem involving $p(x)$-growth structure with a strong dependence on gradients, J. Evol. Equ. 23, 73, 2023.
  • [16] A. Charkaoui, A. Ben-Loghfyry and S. Zeng, Nonlinear Parabolic Double Phase Variable Exponent Systems with Applications in Image Noise Removal, Applied Mathematical Modelling, 132, 495–530, 2024.
  • [17] A. Charkaoui, H. Fahim and N. E. Alaa, Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent, Opuscula Math. 41, 25–53, 2021.
  • [18] Y. Chen, S. Levineand and M. Rao, Variable exponent linear growth functionals in image restoration, SIAM J. Appl. Math. 66, 1383–1406, 2006.
  • [19] M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal. 215, 443–496, 2015.
  • [20] M. Colombo and G. Mingione, Bounded minimisers of double phase variational integrals, Arch. Ration. Mech. Anal. 218, 219–273, 2015.
  • [21] A. Crespo-Blanco, L. Gasinski, P. Harjulehto and P. Winkert, A new class of double phase variable exponent problems: existence and uniqueness, J. Differerential Equ. 323, 182–228, 2022.
  • [22] C. De Filippis and G. Mingione, Regularity for double phase problems at nearly linear growth, Arch. Ration. Mecha. Anal. 247, 50 pp. 2023.
  • [23] A. El Khalil, P. Lindqvist and A. Touzani, On the stability of the first eigenvalue of the problem: $A_{p}u+\lambda g(x)|u|^{p-2}u=0$ with varying p, Rend. Mat. 24, 321–336, 2004.
  • [24] H. Fahim, A. Charkaoui and N. E. Alaa, Parabolic systems driven by general differential operators with variable exponents and strong nonlinearities with respect to the gradient. J Elliptic Parabol. Equ. 7, 199–219, 2021.
  • [25] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263, 424–446, 2001.
  • [26] G. Fragnelli, Positive periodic solutions for a system of anisotropic parabolic equations, J. Math. Anal. Appl. 367, 204–228, 2010.
  • [27] L. Gasinski and P. Winkert, Existence and uniqueness results for double phase problems with convection term, J. Differ. Equ. 268, 4183–4193, 2020.
  • [28] K. Ho and P. Winkert, New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems, Calc. Var. PDEs 62, 38 pp. 2023.
  • [29] I. H. Kim, Y. H. Kim, M. W. Oh and S. Zeng, Existence and multiplicity of solutions to concave-convex-type double-phase problems with variable exponent, Nonlinear Anal. 67, 103627, 2022.
  • [30] O. Kovácik and J. Rákosník, On spaces $L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$, Czechoslovak Math. J. 41, 592–618, 1991.
  • [31] W. Liu and G. Dai, Existence and multiplicity results for double phase problem, J. Differ. Equ. 265, 4311–4334, 2018.
  • [32] W. Liu and G. Dai, Three ground state solutions for double phase problem, J. Math. Phys. 59, 121503, 2018.
  • [33] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with non standard growth conditions, Arch. Ration. Mech. Anal. 105, 267–284, 1989.
  • [34] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, qgrowth conditions, J. Differ. Equ. 90, 1–30, 1991.
  • [35] M. Mihˇailescu and V. Rˇadulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A 462, 2625–2641, 2006.
  • [36] M. Mihˇailescu and D. Repovš, On a PDE involving the $\mathcal{A}_{p(\cdot)}$-Laplace operator, Nonlinear Anal. 75, 975–981, 2012.
  • [37] N. S. Papageorgiou, V. D. Radulescu and D. D. Repov˘s, Double-phase problems and a discontinuity property of the spectrum, P. Am. Math. Soc. 147, 2899–2910, 2019.
  • [38] V. D. Radulescu and D. D. Repov˘s, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press Taylor and Francis Group, 2015.
  • [39] K. Rajagopal, Mathematical modelling of electrorheological fluids, Contin. Mech. Thermodyn. 13, 59–78, 2001.
  • [40] Y. G. Reshetnyak, Set of singular points of solutions of certain nonlinear elliptic equations, Sibirsk. Mat. Zh. 9, 354–368, 1968. (in Russian).
  • [41] M. Ruzicka, Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science and Business Media, 2000.
  • [42] S. Zeng, N. S. Papageorgiou and P. Winkert, Inverse problems for double-phase obstacle problems with variable exponents, J. Optim. Theory Appl. 196, 666–699, 2023.
  • [43] S. Zeng, V. D. Radulescu and P. Winkert, Double phase obstacle problems with variable exponent, Adv. Differential Equ. 27, 611–645, 2022.
  • [44] S. Zeng, V. D. Radulescu and P. Winkert, Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions, SIAM J. Math. Anal. 54, 1898–1926, 2022.
  • [45] Q. Zhang and V. D. Rˇadulescu, Double phase anisotropic variational problems and combined effects of reaction and absorption terms, J. Math. Pures Appl. 118, 159–203, 2018.
  • [46] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Izv. Ros. Akadem. Nauk. 50, 675–710, 1986.
  • [47] V. V. Zhikov, On Lavrentiev’s phenomenon, Russian J. Math. Phy. 3, 2 pp. 1995.
  • [48] V. V. Zhikov, On some variational problems, Russian J. Math. Phys. 5, 105 pp. 1997.
There are 48 citations in total.

Details

Primary Language English
Subjects Partial Differential Equations, Operator Algebras and Functional Analysis
Journal Section Mathematics
Authors

Abderrahim Charkaoui 0000-0003-1425-7248

Jinlan Pan 0009-0009-6083-3008

Early Pub Date August 27, 2024
Publication Date April 28, 2025
Submission Date February 2, 2024
Acceptance Date May 14, 2024
Published in Issue Year 2025 Volume: 54 Issue: 2

Cite

APA Charkaoui, A., & Pan, J. (2025). Double phase variable exponent problems with nonlinear matrices diffusion. Hacettepe Journal of Mathematics and Statistics, 54(2), 445-456. https://doi.org/10.15672/hujms.1428174
AMA Charkaoui A, Pan J. Double phase variable exponent problems with nonlinear matrices diffusion. Hacettepe Journal of Mathematics and Statistics. April 2025;54(2):445-456. doi:10.15672/hujms.1428174
Chicago Charkaoui, Abderrahim, and Jinlan Pan. “Double Phase Variable Exponent Problems With Nonlinear Matrices Diffusion”. Hacettepe Journal of Mathematics and Statistics 54, no. 2 (April 2025): 445-56. https://doi.org/10.15672/hujms.1428174.
EndNote Charkaoui A, Pan J (April 1, 2025) Double phase variable exponent problems with nonlinear matrices diffusion. Hacettepe Journal of Mathematics and Statistics 54 2 445–456.
IEEE A. Charkaoui and J. Pan, “Double phase variable exponent problems with nonlinear matrices diffusion”, Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, pp. 445–456, 2025, doi: 10.15672/hujms.1428174.
ISNAD Charkaoui, Abderrahim - Pan, Jinlan. “Double Phase Variable Exponent Problems With Nonlinear Matrices Diffusion”. Hacettepe Journal of Mathematics and Statistics 54/2 (April 2025), 445-456. https://doi.org/10.15672/hujms.1428174.
JAMA Charkaoui A, Pan J. Double phase variable exponent problems with nonlinear matrices diffusion. Hacettepe Journal of Mathematics and Statistics. 2025;54:445–456.
MLA Charkaoui, Abderrahim and Jinlan Pan. “Double Phase Variable Exponent Problems With Nonlinear Matrices Diffusion”. Hacettepe Journal of Mathematics and Statistics, vol. 54, no. 2, 2025, pp. 445-56, doi:10.15672/hujms.1428174.
Vancouver Charkaoui A, Pan J. Double phase variable exponent problems with nonlinear matrices diffusion. Hacettepe Journal of Mathematics and Statistics. 2025;54(2):445-56.

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