Local $ L^\infty $ norm estimates for the gradient solutions of variational inequalities arising from the mortgage problems

Qingjun Zhao; Qingjun Zhao

doi:10.3934/math.2025673

AIMS Mathematics

2025, Volume 10, Issue 6: 15012-15024. doi: 10.3934/math.2025673

Previous Article Next Article

Research article

Local $ L^\infty $ norm estimates for the gradient solutions of variational inequalities arising from the mortgage problems

Qingjun Zhao ^,

School of Economics and Management, Chongqing Normal University, Chongqing 401331, China

Received: 24 April 2025 Revised: 18 June 2025 Accepted: 24 June 2025 Published: 30 June 2025
MSC : 35K99, 97M30

This paper investigates local estimates for the spatial gradient of solutions to variational inequalities within the framework of a parabolic Kirchhoff operator, which arises from mortgage problems. By utilizing the integral inequality for the gradient of the solutions derived in this study, together with the Caffarelli–Kohn–Nirenberg inequality, we establish an $ L^\infty $ norm estimate for the gradient of the solution in a local cylindrical region. This $ L^\infty $ estimate is formulated in terms of the $ L^p $ norm of the solution.
Citation: Qingjun Zhao. Local $ L^\infty $ norm estimates for the gradient solutions of variational inequalities arising from the mortgage problems[J]. AIMS Mathematics, 2025, 10(6): 15012-15024. doi: 10.3934/math.2025673

Related Papers:

Abstract

This paper investigates local estimates for the spatial gradient of solutions to variational inequalities within the framework of a parabolic Kirchhoff operator, which arises from mortgage problems. By utilizing the integral inequality for the gradient of the solutions derived in this study, together with the Caffarelli–Kohn–Nirenberg inequality, we establish an $ L^\infty $ norm estimate for the gradient of the solution in a local cylindrical region. This $ L^\infty $ estimate is formulated in terms of the $ L^p $ norm of the solution.

References

[1]	X. He, P. Pasricha, S. Lin, Analytically pricing European options in dynamic markets: Incorporating liquidity variations and economic cycles, Econ. Model., 139 (2024), 106839. https://doi.org/10.1016/j.econmod.2024.106839 doi: 10.1016/j.econmod.2024.106839
[2]	W. Chen, S. Wang, A penalty method for a fractional order parabolic variational inequality governing American put option valuation, Comput. Math. Appl., 67 (2014), 77–90. https://doi.org/10.1016/j.camwa.2013.10.007 doi: 10.1016/j.camwa.2013.10.007
[3]	Z. Peng, Y. Zhao, F. Long, Existence and uniqueness of the solution to a new class of evolutionary variational hemivariational inequalities, Nonlinear Anal.-Real, 81 (2025), 104210. https://doi.org/10.1016/j.nonrwa.2024.104210 doi: 10.1016/j.nonrwa.2024.104210
[4]	S. Treant$\mathrm{\breve{a}}$, V. Singh, S. K. Mishra, Reciprocal solution existence results for a class of vector variational control inequalities with application in physics, J. Comput. Appl. Math., 461 (2025), 116461. https://doi.org/10.1016/j.cam.2024.116461 doi: 10.1016/j.cam.2024.116461
[5]	Z. Zhang, X. Nie, J. Cao, Variational inequalities of multilayer elastic contact systems with interlayer friction: Existence and uniqueness of solution and convergence of numerical solution, Comput. Math. Appl., 174 (2024), 248–260. https://doi.org/10.1016/j.camwa.2024.08.030 doi: 10.1016/j.camwa.2024.08.030
[6]	Z. Wu, W. Li, Q. Zhang, Y. Xiao, New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities, J. Comput. Appl. Math., 448 (2024), 115926. https://doi.org/10.1016/j.cam.2024.115926 doi: 10.1016/j.cam.2024.115926
[7]	D. C. Antonopoulou, G. Dewhirst, G. Karali, K. Tzirakis, Local existence of the outer parabolic stochastic Stefan problem on the sphere, J. Differ. Equations, 423 (2025), 439–475. https://doi.org/10.1016/j.jde.2025.01.005 doi: 10.1016/j.jde.2025.01.005
[8]	N. Chen, F. Li, P. Wang, Global existence and blow-up phenomenon for a nonlocal semilinear pseudo-parabolic p-Laplacian equation, J. Differ. Equations, 409 (2024), 395–440. https://doi.org/10.1016/j.jde.2024.07.008 doi: 10.1016/j.jde.2024.07.008
[9]	J. Wang, X. Wang, Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity, Nonlinear Anal.-Real, 81 (2025), 104220. https://doi.org/10.1016/j.nonrwa.2024.104220 doi: 10.1016/j.nonrwa.2024.104220
[10]	K. Zhang, F. Dong, A priori estimates and existence of positive stationary solutions for semilinear parabolic equations, J. Math. Anal. Appl., 542 (2025), 128770. https://doi.org/10.1016/j.jmaa.2024.128770 doi: 10.1016/j.jmaa.2024.128770
[11]	X. Li, $W ^{2, \delta }$ estimates for fully nonlinear parabolic inequalities on $C^{1, \alpha }$ domains, Appl. Math. Lett., 163 (2025), 109459. https://doi.org/10.1016/j.aml.2025.109459 doi: 10.1016/j.aml.2025.109459
[12]	D. Li, X. Li, $W^{2, \delta}$ estimates for solution sets of fully nonlinear elliptic inequalities on $C^{1, \alpha}$ domains, J. Funct. Anal., 285 (2023), 109985. https://doi.org/10.1016/j.jfa.2023.109985 doi: 10.1016/j.jfa.2023.109985
[13]	A. Bui, Maximal regularity of parabolic equations associated with a discrete Laplacian, J. Differ. Equations, 375 (2023), 277–303. https://doi.org/10.1016/j.jde.2023.07.043 doi: 10.1016/j.jde.2023.07.043
[14]	S. Das, Gradient Hölder regularity in mixed local and nonlocal linear parabolic problem, J. Math. Anal. Appl., 535 (2024), 128140. https://doi.org/10.1016/j.jmaa.2024.128140 doi: 10.1016/j.jmaa.2024.128140
[15]	L. Wang, Gradient estimates on the weighted p-Laplace heat equation, J. Differ. Equations, 264 (2018), 506–524. https://doi.org/10.1016/j.jde.2017.09.012 doi: 10.1016/j.jde.2017.09.012
[16]	Y. Sun, T. Wu, Hölder and Schauder estimates for weak solutions of a certain class of non-divergent variation inequality problems in finance, AIMS Math., 8 (2023), 18995–19003. https://doi.org/10.3934/math.2023968 doi: 10.3934/math.2023968
[17]	W. Chen, T. Zhou, Existence of solutions for p-Laplacian parabolic Kirchhoff equation, Appl. Math. Lett., 122 (2021), 107527. https://doi.org/10.1016/j.aml.2021.107527 doi: 10.1016/j.aml.2021.107527
[18]	F. Liu, M. Nadeem, I. Mahariq, S. Dawood, A numerical approach for the analytical solution of the fourth-order parabolic partial differential equations, J. Funct. Space., 2022, 3309674. https://doi.org/10.1155/2022/3309674
[19]	M. Nadeem, Q. Ain, Y. Alsayaad, Approximate solutions of multidimensional wave problems using an effective approach, J. Funct. Space., 2023, 5484241. https://doi.org/10.1155/2023/5484241
[20]	E. Henriques, Local Hölder continuity to a class of doubly singular nonlinear evolutionary equations, J. Math. Anal. Appl., 518 (2023), 126676. https://doi.org/10.1016/j.jmaa.2022.126676 doi: 10.1016/j.jmaa.2022.126676

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)