具有部分BMO系数的非散度型抛物方程的Lorentz估计

数学物理学报 ›› 2019, Vol. 39 ›› Issue (6): 1405-1420.

具有部分BMO系数的非散度型抛物方程的Lorentz估计

张俊杰¹(),郑神州²(),于海燕³()

¹ 河北师范大学数学科学学院石家庄 050024
² 北京交通大学理学院北京 100044
³ 内蒙古民族大学数学学院内蒙古通辽 028043

收稿日期:2018-06-28 出版日期:2019-12-26 发布日期:2019-12-28
作者简介:张俊杰, E-mail: junjiezhang@hebtu.edu.cn|郑神州, E-mail: shzhzheng@bjtu.edu.cn|于海燕, E-mail: jiechy@163.com
基金资助:
河北师范大学科研基金(L2019B02);河北省自然科学基金(A2019205218);内蒙古自治区自然科学基金(2018MS01008);内蒙古自治区高等学校科学研究项目(NJZY18164)

Lorentz Estimates for Nondivergence Parabolic Equations with Partially BMO Coefficients

Junjie Zhang¹(),Shenzhou Zheng²(),Haiyan Yu³()

¹ College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050016
² Department of Mathematics, Beijing Jiaotong University, Beijing 100044
³ College of Mathematics, Inner Mongolia University for Nationalities, Inner Mongolia Tongliao 028043

Received:2018-06-28 Online:2019-12-26 Published:2019-12-28
Supported by:
the Science Foundation of Hebei Normal University(L2019B02);the Natural Science Foundation of Hebei Province(A2019205218);the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region(2018MS01008);the Science Research Program of Institution of Higher Education at Universities of Inner Mongolia Autonomous Region(NJZY18164)

摘要/Abstract

摘要：

该文利用"大M不等式原理"证明了非散度型线性抛物方程

$ {{u}_{t}}-{{a}_{ij}}(x,t){{D}_{ij}}u(x,t)=f(x,t) $

强解Hessian矩阵的内部Lorentz估计，其中主项系数a_ij（x，t）满足一致抛物条件和部分BMO条件，即a_ij（x，t）关于一个空间变量可测且关于其余变量具有小的BMO半范数.

关键词: 非散度型抛物方程, Lorentz空间, 部分BMO

Abstract:

In this paer, we prove an interior Lorentz estimate for Hessian of the strong solutions to nondivergence linear parabolic equations u_t -a_ij(x, t)D_iju(x, t)=f(x, t). Here, the leading coefficients a_ij(x, t) are assumed to be merely measurable in one spatial variable and have small BMO semi-norms with respect to the remaining variables.

Key words: Nondivergence parabolic equations, Lorentz spaces, Partially BMO

中图分类号:

O175.2

张俊杰,郑神州,于海燕. 具有部分BMO系数的非散度型抛物方程的Lorentz估计[J]. 数学物理学报, 2019, 39(6): 1405-1420.

Junjie Zhang,Shenzhou Zheng,Haiyan Yu. Lorentz Estimates for Nondivergence Parabolic Equations with Partially BMO Coefficients[J]. Acta mathematica scientia,Series A, 2019, 39(6): 1405-1420.

参考文献

1	Acerbi E Mingione G Gradient estimates for a class of parabolic systems Duke Math J 2007, 136 (2): 285- 320
2	Baroni P Lorentz estimates for degenerate and singular evolutionary systems J Diff Equ 2013, 255: 2927- 2951
3	Bramanti M Cerutti M C W_p^{1, 2}-solvability for the Cauchy-Dirichlet problem for parabolic equations with VMO coefficients Comm Part Diff Equ 1993, 18 (9/10): 1735- 1763
4	Byun S S Lee M On weighted W_p^{1, 2}-estimates for elliptic equations with BMO coefficients in nondivergence form International J Math 2015, 26 (1): 1550001
5	Byun S S Lee K A Oh J Park J Nondivergence elliptic and parabolic problems with irregular obstacles Math Z 2018, 290 (3/4): 973- 990
6	Chiarenza F Frasca M Longo P Interior W^{2, p}-estimates for nondivergence elliptic equations with discontinuous coeffcients Ricerche Mat 1991, 40: 149- 168
7	Chiarenza F Frasca M Longo P W^{2, p}-solvability of the Dirichlet problem for nonlinear elliptic equations with VMO coeffcients Trans Amer Math Soc 1993, 336 (2): 841- 853
8	Calderon A P Zygmund A On the existence of certain singular integrals Acta Mathematica 1952, 88 (1): 85- 139
9	Kim D Krylov N V Elliptic differenrial equations with coefficients measurable with respact to one variable and VMO with respect to the others SIAM J Math Anal 2007, 32 (9): 489- 506
10	Kim D Krylov N V Parabolic equations with measurable coefficients Potential Anal 2007, 26 (4): 345- 361
11	Kim D Parabolic equations with measurable coefficients Ⅱ J Math Anal Appl 2007, 334: 534- 548
12	Krylov N V Parabolic and elliptic equations with VMO coeffficients Comm Part Diff Equ 2007, 32: 453- 475
13	Lieberman G M A mostly elementary proof of Morrey space estimates for elliptic and parabolic equations with VMO coefficients J Funct Anal 2003, 201: 457- 479
14	Mengesha T Phuc N C Global estimates for quasilinear elliptic equations on Reifenberg flat domains Arch Rational Mech Anal 2012, 203: 189- 216
15	Mingione G The Calderón-Zygmund theory for elliptic problems with measure data Ann Sc Norm Super Pisa Cl Sci 2007, 6 (5): 195- 261
16	Wang L H A geometric approach to the Calderón-Zygmmund estimates Acta Math Sin 2003, 19: 381- 396
17	Zhang J J Zheng S Z Weighted Lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients Commun Pure Appl Anal 2017, 16 (3): 899- 914
18	Zhang C Zhou S L Global weighted estimates for quasilinear elliptic equations with non-standard growth J Funct Anal 2014, 267: 605- 642

[1]	杨奇祥, 程正兴, 彭立中. 小波与函数空间[J]. 数学物理学报, 2005, 25(1): 130-144.
[2]	匡继昌. 多维加权特殊原子空间上的Riesz平均算子[J]. 数学物理学报, 1998, 18(1): 6-10.
[3]	刘岚喆. 齐型空间上的Poisson算子与广义Carleson测度[J]. 数学物理学报, 1997, 17(2): 200-206.