-
Previous Article
Riesz-type representation formulas for subharmonic functions in sub-Riemannian settings
- CPAA Home
- This Issue
-
Next Article
Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs
$ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients
1. | School of Mathematical Sciences, Peking Uinversity, Beijing 100871, China |
2. | School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an 710049, China |
In this paper, we establish uniform $ W^{1,p} $ estimates for composite material problems which can be described by a divergence form elliptic system on a nonsmooth domain composed of a finite number of subdomains. We want to derive global $ W^{1,p} $ regularity under the assumption that the coefficients are almost $ (\delta,R) $-vanishing of codimension 1 (see Definition 1.2) in each of multiple subdomains and the boundaries of subdomains are Reifenberg flat, moreover the estimates do not depend on the distance between these subdomains.
References:
[1] |
S. Byun, S. Ryu and L. Wang,
Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains, Manuscripta Math., 133 (2010), 225-245.
doi: 10.1007/s00229-010-0373-1. |
[2] |
S. Byun and L. Wang,
Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310.
doi: 10.1002/cpa.20037. |
[3] |
L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, (1995).
doi: 10.1090/coll/043. |
[4] |
M. Chipot, D. Kinderlehrer and G. V. Caffarelli,
Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96.
doi: 10.1007/BF00251414. |
[5] |
Y. Li and M. Vogelius,
Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients (English summary), Arch. Ration. Mech. Anal., 153 (2000), 91-151.
doi: 10.1007/s002050000082. |
[6] |
Y. Li and L. Nirenberg,
Estimates for elliptic systems from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.
doi: 10.1002/cpa.10079. |
[7] |
E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, (1993).
doi: 10.1515/9781400883929. |
[8] |
K. Um,
Elliptic equations with singular BMO coefficients in Reifenberg domains, J. Differ. Equ., 253 (2012), 2993-3015.
doi: 10.1016/j.jde.2012.08.016. |
show all references
References:
[1] |
S. Byun, S. Ryu and L. Wang,
Gradient estimates for elliptic systems with measurable coefficients in nonsmooth domains, Manuscripta Math., 133 (2010), 225-245.
doi: 10.1007/s00229-010-0373-1. |
[2] |
S. Byun and L. Wang,
Elliptic equations with BMO coefficients in Reifenberg domains, Commun. Pure Appl. Math., 57 (2004), 1283-1310.
doi: 10.1002/cpa.20037. |
[3] |
L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43. American Mathematical Society, Providence, RI, (1995).
doi: 10.1090/coll/043. |
[4] |
M. Chipot, D. Kinderlehrer and G. V. Caffarelli,
Smoothness of linear laminates, Arch. Rational Mech. Anal., 96 (1986), 81-96.
doi: 10.1007/BF00251414. |
[5] |
Y. Li and M. Vogelius,
Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients (English summary), Arch. Ration. Mech. Anal., 153 (2000), 91-151.
doi: 10.1007/s002050000082. |
[6] |
Y. Li and L. Nirenberg,
Estimates for elliptic systems from composite material, Commun. Pure Appl. Math., 56 (2003), 892-925.
doi: 10.1002/cpa.10079. |
[7] |
E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, (1993).
doi: 10.1515/9781400883929. |
[8] |
K. Um,
Elliptic equations with singular BMO coefficients in Reifenberg domains, J. Differ. Equ., 253 (2012), 2993-3015.
doi: 10.1016/j.jde.2012.08.016. |
[1] |
Byungsoo Kang, Hyunseok Kim. W1, p-estimates for elliptic equations with lower order terms. Communications on Pure and Applied Analysis, 2017, 16 (3) : 799-822. doi: 10.3934/cpaa.2017038 |
[2] |
Lele Du. Bounds for subcritical best Sobolev constants in W1, p. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3871-3886. doi: 10.3934/cpaa.2021135 |
[3] |
Bojing Shi. $ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 537-553. doi: 10.3934/dcds.2021127 |
[4] |
Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141 |
[5] |
N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073 |
[6] |
Yi Cao, Dong Li, Lihe Wang. The optimal weighted $W^{2, p}$ estimates of elliptic equation with non-compatible conditions. Communications on Pure and Applied Analysis, 2011, 10 (2) : 561-570. doi: 10.3934/cpaa.2011.10.561 |
[7] |
Alexandre N. Carvalho, Jan W. Cholewa. Strongly damped wave equations in $W^(1,p)_0 (\Omega) \times L^p(\Omega)$. Conference Publications, 2007, 2007 (Special) : 230-239. doi: 10.3934/proc.2007.2007.230 |
[8] |
Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213 |
[9] |
Sofia Giuffrè, Giovanna Idone. On linear and nonlinear elliptic boundary value problems in the plane with discontinuous coefficients. Discrete and Continuous Dynamical Systems, 2011, 31 (4) : 1347-1363. doi: 10.3934/dcds.2011.31.1347 |
[10] |
Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure and Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751 |
[11] |
Sun-Sig Byun, Lihe Wang. $W^{1,p}$ regularity for the conormal derivative problem with parabolic BMO nonlinearity in reifenberg domains. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 617-637. doi: 10.3934/dcds.2008.20.617 |
[12] |
Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080 |
[13] |
Jean Dolbeault, Marta García-Huidobro, Rául Manásevich. Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 375-394. doi: 10.3934/dcds.2020014 |
[14] |
Giorgio Metafune, Chiara Spina. Heat Kernel estimates for some elliptic operators with unbounded diffusion coefficients. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2285-2299. doi: 10.3934/dcds.2012.32.2285 |
[15] |
Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure and Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043 |
[16] |
Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721 |
[17] |
Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315 |
[18] |
Horst Heck, Matthias Hieber, Kyriakos Stavrakidis. $L^\infty$-estimates for parabolic systems with VMO-coefficients. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 299-309. doi: 10.3934/dcdss.2010.3.299 |
[19] |
Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure and Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647 |
[20] |
Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial and Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]