September  2021, 20(9): 3143-3159. doi: 10.3934/cpaa.2021100

$ 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

* Corresponding author

Received  April 2018 Revised  May 2021 Published  September 2021 Early access  June 2021

Fund Project: The first author is supported by NSF grant 8206400077

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.

Citation: Caiyan Li, Dongsheng Li. $ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3143-3159. doi: 10.3934/cpaa.2021100
References:
[1]

S. ByunS. 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. ChipotD. 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. ByunS. 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. ChipotD. 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

Metrics

  • PDF downloads (141)
  • HTML views (163)
  • Cited by (0)

Other articles
by authors

[Back to Top]