BẤT ĐẲNG THỨC PHÂN PHỐI CHO MỘT LỚP CÁC PHƯƠNG TRÌNH ELLIPTIC TỰA TUYẾN TÍNH VỚI DỮ LIỆU HỖN HỢP
Nội dung chính của bài viết
Tóm tắt
Bài toán về tính chính quy nghiệm cho các phương trình đạo hàm riêng đã được nhiều nhà toán học nghiên cứu trong những năm gần đây bằng nhiều phương pháp khác nhau. Với sự phát triển của giải tích điều hòa, lí thuyết Calderón-Zygmund đóng vai trò quan trọng trong việc nghiên cứu bài toán chính quy nghiệm. Trong bài báo này, chúng tôi thiết lập đánh giá dạng Calderón-Zygmund cho nghiệm yếu của một lớp phương trình elliptic tựa tuyến tính với dữ liệu hỗn hợp trong không gian Lorentz tổng quát. Nghiên cứu của chúng tôi là một dạng mở rộng liên quan đến không gian hàm đối với một số đánh giá gradient trong một số bài báo mới đây. Kết quả này một lần nữa khẳng định tính hiệu quả của phương pháp sử dụng bất đẳng thức phân phối trên các tập mức đối với bài toán chính quy nghiệm cho phương trình đạo hàm riêng.
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Tài liệu tham khảo
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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.
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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.