GLOBAL GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC EQUATIONS

Title & Authors
GLOBAL GRADIENT ESTIMATES FOR NONLINEAR ELLIPTIC EQUATIONS
Ryu, Seungjin;

Abstract
We prove global gradient estimates in weighted Orlicz spaces for weak solutions of nonlinear elliptic equations in divergence form over a bounded non-smooth domain as a generalization of Calder$\small{\acute{o}}$n-Zygmund theory. For each point and each small scale, the main assumptions are that nonlinearity is assumed to have a uniformly small mean oscillation and that the boundary of the domain is sufficiently flat.
Keywords
nonlinear elliptic equation;global gradient estimate;Calder$\small{\acute{o}}$n-Zygmund theory;BMO space;Reifenberg flat domain;
Language
English
Cited by
1.
Weighed Estimates for Nonlinear Elliptic Problems with Orlicz Data, Journal of Elliptic and Parabolic Equations, 2015, 1, 1, 49
References
1.
E. Acerbi and G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J. 136 (2007), no. 2, 285-320.

2.
P. Baroni, Lorentz estimates for degenerate and singular evolutionary systems, J. Differential Equations 255 (2013), no. 9, 2927-2951.

3.
P. Baroni, A. Di Castro, and G. Palatucci, Global estimates for nonlinear parabolic equations, J. Evol. Equ. 13 (2013), no. 1, 163-195.

4.
S. Byun, J. Ok, and S. Ryu, Global gradient estimates for general nonlinear parabolic equations in nonsmooth domains, J. Differential Equations 254 (2013), no. 11, 4290-4326.

5.
S. Byun and D. K. Palagachev, Weighted \$L^{P}\$-estimates for elliptic equations with mea- surable coefficients in nonsmooth domains, Potential Anal. 41 (2014), no. 1, 51-79.

6.
S. Byun, D. K. Palagachev, and S. Ryu, Weighted \$W^{1,P}\$ estimates for solutions of non-linear parabolic equations over non-smooth domains, Bull. London Math. Soc. 45 (2013), no. 4, 765-778.

7.
S. Byun and S. Ryu, Gradient estimates for higher order elliptic equations on nonsmooth domains, J. Differential Equations 250 (2011), no. 1, 243-263.

8.
S. Byun and S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations, Ann. Inst. H. Poincare Anal. Non Lineaire 30 (2013), no. 2, 291-313.

9.
S. Byun and L. Wang, Elliptic equations with BMO nonlinearity in Reifenberg domains, Adv. Math. 219 (2008), no. 6, 1937-1971.

10.
L. A. Caffarelli and I. Peral, On \$W^{1,P}\$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998), no. 1, 1-21.

11.
H. Dong and D. Kim, Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains, J. Funct. Anal. 261 (2011), no. 11, 3279-3327.

12.
H. Dong and D. Kim, \$L_{P}\$ solvability of divergence type parabolic and elliptic systems with partially BMO coefficients, Calc. Var. Partial Differential Equations 40 (2011), no. 3-4, 357-389.

13.
H. Dong and D. Kim, The conormal derivative problem for higher order elliptic systems with irregular coefficients, Recent advances in harmonic analysis and partial differential equations, 69-97, Contemp. Math., 581, Amer. Math. Soc., Providence, RI, 2012.

14.
F. Duzaar, G. Mingione, and K. Steffen, Parabolic systems with polynomial growth and regularity, Mem. Amer. Math. Soc. 214 (2011), no. 1005, x+118 pp.

15.
G. Di Fazio, \$L^{P}\$ estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Un. Mat. Ital A (7) 10 (1996), no. 2, 409-420.

16.
A. Fiorenza and M. Krbec, Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolin. 38 (1997), no. 3, 433-451.

17.
V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific Publishing Co., Inc., River Edge, NJ, 1991.

18.
N. V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (2007), no. 1-3, 453-475.

19.
N. V. Krylov, Lectures on elliptic and parabolic equations in Sobolev spaces, Graduate Studies in Mathematics, 96. American Mathematical Society, Providence, RI, 2008. xviii+357pp.

20.
T. Mengesha and N. C. Phuc, Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains, J. Differential Equations 250 (2011), no. 5, 2485-2507.

21.
T. Mengesha and N. C. Phuc, Global Estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal. 203 (2012), no. 1, 189-216.

22.
D. K. Palagachev and L. G. Softova, A priori estimates and precise regularity for parabolic systems with discontinuous data, Discrete Contin. Dyn. Syst. 13 (2005), no. 3, 721-742.

23.
N. C. Phuc, Nonlinear Muckenhoupt-Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations, Adv. Math. 250 (2014), 387-419.

24.
L. G. Softova, Morrey-type regularity of solutions to parabolic problems with discontinuous data, Manuscripta Math. 136 (2011), no. 3-4, 365-382.

25.
T. Toro, Doubling and flatness: geometry of measures, Notices Amer. Math. Soc. 44 (1997), no. 9, 1087-1094.

26.
L. Wang and F. Yao, Global regularity for higher order divergence elliptic and parabolic equations, J. Funct. Anal. 266 (2014), no. 2, 792-813.

27.
L.Wang, F. Yao, S. Zhou, and H. Jia, Optimal regularity theory for the Poisson equation, Proc. Amer. Math. Soc. 137 (2009), no. 6, 2037-2047.