DISTRIBUTION INEQUALITY FOR A CLASS OF QUASI-LINEAR ELLIPTIC EQUATIONS WITH MIXED DATA
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Abstract
The problem of regularity for partial differential equations has been studied by many mathematicians in recent years using many different methods. With the development of harmonic analysis, Calderón-Zygmund theory plays an important role in investigating the regularity problem. In this paper, we establish Calderón-Zygmund type estimate for weak solutions to a class of quasi-linear elliptic equations with mixed data in the generalized Lorentz space. Our study is an extension related to the function space to some gradient estimates in several previous papers. This result once again confirms the effectiveness of the method of using the distribution inequality on the level sets to the regularity problem for partial differential equations.
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References
Breit, D., Cianchi, A., Diening, L., Kuusi, T., & Schwarzacher, S. (2017). The p-Laplace system with right-hand side in divergence form: inner and up to the boundary pointwise estimates. Nonlinear Anal., 153, 200–212.
Byun, S.-S. & Wang, L. (2004). Elliptic equations with BMO coefficients in Reifenberg domains. Comm. Pure Appl. Math., 57, 1283–1310.
Byun, S.-S. & Wang, L. (2008). Elliptic equations with BMO nonlinearity in Reifenberg domains. Adv. Math., 219 (6), 1937–1971.
Caffarelli, L. A. & Peral, I. (1998). On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math., 51 (1), 1–21.
DiBenedetto, E. & Manfredi, J. (1993). On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Amer. J. Math., 115 (5), 1107–1134.
Duzaar, F. & Mingione, G. (2010). Gradient estimates via linear and nonlinear potentials. J. Functional Analysis, 259, 2961–2998.
Duzaar, F. & Mingione, G. (2011). Gradient estimates via non-linear potentials. Amer. J. Math., 133, 1093–1149.
Evans, L. (1982). A new proof of local C1,α regularity for solutions of certain degenerate elliptic PDE. J. Differential Equations, 145, 356–373.
Grafakos, L. (2004). Classical and Modern Fourier Analysis, Pearson/Prentice Hall.
Iwaniec, T. (1983). Projections onto gradient fields and Lp-estimates for degenerated elliptic operators. Stud. Math., 75 (3), 293–312.
Lee, M. & Ok, J. (2019). Nonlinear Calderón-Zygmund theory involving dual data. Rev. Mat. Iberoamericana, 35 (4), 10530–1078.
Lieberman, G. M. (1984). Solvability of quasilinear elliptic equations with nonlinear boundary conditions. J. Functional Analysis, 56 (2), 210–219.
Milakis, E. & Toro, T. (2010). Divergence form operators in Reifenberg flat domains. Math. Z., 264(1), 15–41.
Mingione, G. (2007). The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scuola. Norm. Super. Pisa Cl. Sci. (V), 6, 195–261.
Mingione, G. (2010). Gradient estimates below the duality exponent. Math. Ann., 346, 571–627.
Nguyen, T.-N. & Tran, M.-P. (2020). Lorentz improving estimates for the p-Laplace equations with mixed data. Nonlinear Anal., 200, 111960.
Nguyen, T.-N. & Tran, M.-P. (2021). Level-set inequalities on fractional maximal distribution functions and applications to regularity theory. J. Functional Analysis, 280 (1), 108797.
Nguyen, T.-N., Tran, M.-P., Doan, C.-K., & Vo, V.-N. (2021). A gradient estimate related fractional maximal operators for a p-Laplace problem in Morrey spaces. Taiwanese J. Math., 25 (4), 809–829.
Tolksdorff, P. (1984). Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51 (1), 126–150.
Tran, M.-P. (2019). Good-λ type bounds of quasilinear elliptic equations for the singular case. Nonlinear Anal., 178, 266–281.
Tran, M.-P. & Nguyen, T.-N. (2019). Generalized good-λ techniques and applications to weighted Lorentz regularity for quasilinear elliptic equations. C. R. Math., 357 (8), 664–670.
Tran, M.-P. & Nguyen, T.-N. (2020a). New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data. J. Differential Equations, 268 (4), 1427–1462.
Tran, M.-P. & Nguyen, T.-N. (2020b). Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data. Commun. Contemp. Math., 22 (5), 1950033.
Tran, M.-P. & Nguyen, T.-N. (2022). Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data. Nonlinear Anal., 214, 112613.
Tran, M.-P. & Nguyen, T.-N. (2023). Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications. Nonlinear Anal. Real World Appl., 69, 103750.
Tran, M.-P., Nguyen, T.-N., & Nguyen, H.-N. (2024). Regularity for the steady Stokes-type flow of incompressible Newtonian fluids in some generalized function settings. Nonlinear Anal. Real World Appl., 77, 104049.
Tran, M.-P., Tran, T.-Q., & Nguyen, T.-N. (2024). Global bound on the gradient of solutions to p-Laplace type equations with mixed data. Acta Math. Sci., 44 (4), 1394-1414.
Uhlenbeck, K. (1977). Regularity for a class of nonlinear elliptic systems. Acta. Math., 138 (3-4), 219–240.
Byun, S.-S. & Wang, L. (2004). Elliptic equations with BMO coefficients in Reifenberg domains. Comm. Pure Appl. Math., 57, 1283–1310.
Byun, S.-S. & Wang, L. (2008). Elliptic equations with BMO nonlinearity in Reifenberg domains. Adv. Math., 219 (6), 1937–1971.
Caffarelli, L. A. & Peral, I. (1998). On W 1,p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math., 51 (1), 1–21.
DiBenedetto, E. & Manfredi, J. (1993). On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Amer. J. Math., 115 (5), 1107–1134.
Duzaar, F. & Mingione, G. (2010). Gradient estimates via linear and nonlinear potentials. J. Functional Analysis, 259, 2961–2998.
Duzaar, F. & Mingione, G. (2011). Gradient estimates via non-linear potentials. Amer. J. Math., 133, 1093–1149.
Evans, L. (1982). A new proof of local C1,α regularity for solutions of certain degenerate elliptic PDE. J. Differential Equations, 145, 356–373.
Grafakos, L. (2004). Classical and Modern Fourier Analysis, Pearson/Prentice Hall.
Iwaniec, T. (1983). Projections onto gradient fields and Lp-estimates for degenerated elliptic operators. Stud. Math., 75 (3), 293–312.
Lee, M. & Ok, J. (2019). Nonlinear Calderón-Zygmund theory involving dual data. Rev. Mat. Iberoamericana, 35 (4), 10530–1078.
Lieberman, G. M. (1984). Solvability of quasilinear elliptic equations with nonlinear boundary conditions. J. Functional Analysis, 56 (2), 210–219.
Milakis, E. & Toro, T. (2010). Divergence form operators in Reifenberg flat domains. Math. Z., 264(1), 15–41.
Mingione, G. (2007). The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Scuola. Norm. Super. Pisa Cl. Sci. (V), 6, 195–261.
Mingione, G. (2010). Gradient estimates below the duality exponent. Math. Ann., 346, 571–627.
Nguyen, T.-N. & Tran, M.-P. (2020). Lorentz improving estimates for the p-Laplace equations with mixed data. Nonlinear Anal., 200, 111960.
Nguyen, T.-N. & Tran, M.-P. (2021). Level-set inequalities on fractional maximal distribution functions and applications to regularity theory. J. Functional Analysis, 280 (1), 108797.
Nguyen, T.-N., Tran, M.-P., Doan, C.-K., & Vo, V.-N. (2021). A gradient estimate related fractional maximal operators for a p-Laplace problem in Morrey spaces. Taiwanese J. Math., 25 (4), 809–829.
Tolksdorff, P. (1984). Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations, 51 (1), 126–150.
Tran, M.-P. (2019). Good-λ type bounds of quasilinear elliptic equations for the singular case. Nonlinear Anal., 178, 266–281.
Tran, M.-P. & Nguyen, T.-N. (2019). Generalized good-λ techniques and applications to weighted Lorentz regularity for quasilinear elliptic equations. C. R. Math., 357 (8), 664–670.
Tran, M.-P. & Nguyen, T.-N. (2020a). New gradient estimates for solutions to quasilinear divergence form elliptic equations with general Dirichlet boundary data. J. Differential Equations, 268 (4), 1427–1462.
Tran, M.-P. & Nguyen, T.-N. (2020b). Lorentz-Morrey global bounds for singular quasilinear elliptic equations with measure data. Commun. Contemp. Math., 22 (5), 1950033.
Tran, M.-P. & Nguyen, T.-N. (2022). Global gradient estimates for very singular quasilinear elliptic equations with non-divergence data. Nonlinear Anal., 214, 112613.
Tran, M.-P. & Nguyen, T.-N. (2023). Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications. Nonlinear Anal. Real World Appl., 69, 103750.
Tran, M.-P., Nguyen, T.-N., & Nguyen, H.-N. (2024). Regularity for the steady Stokes-type flow of incompressible Newtonian fluids in some generalized function settings. Nonlinear Anal. Real World Appl., 77, 104049.
Tran, M.-P., Tran, T.-Q., & Nguyen, T.-N. (2024). Global bound on the gradient of solutions to p-Laplace type equations with mixed data. Acta Math. Sci., 44 (4), 1394-1414.
Uhlenbeck, K. (1977). Regularity for a class of nonlinear elliptic systems. Acta. Math., 138 (3-4), 219–240.