Quasilinear elliptic equations with BMO coefficients in Lipschitz domains
HTML articles powered by AMS MathViewer
- by Sun-Sig Byun and Lihe Wang PDF
- Trans. Amer. Math. Soc. 359 (2007), 5899-5913 Request permission
Abstract:
We obtain a global $W^{1,q}$ estimate for the weak solution to an elliptic partial differential equation of $p$-Laplacian type with BMO coefficients in a Lipschitz domain with small Lipschitz constant.References
- Sun-Sig Byun, Elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc. 357 (2005), no. 3, 1025–1046. MR 2110431, DOI 10.1090/S0002-9947-04-03624-4
- Luis A. Caffarelli and Xavier Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1351007, DOI 10.1090/coll/043
- 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. MR 1486629, DOI 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N
- E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math. 115 (1993), no. 5, 1107–1134. MR 1246185, DOI 10.2307/2375066
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Tadeusz Iwaniec, Projections onto gradient fields and $L^{p}$-estimates for degenerated elliptic operators, Studia Math. 75 (1983), no. 3, 293–312. MR 722254, DOI 10.4064/sm-75-3-293-312
- David Jerison and Carlos E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1995), no. 1, 161–219. MR 1331981, DOI 10.1006/jfan.1995.1067
- David S. Jerison and Carlos E. Kenig, The logarithm of the Poisson kernel of a $C^{1}$ domain has vanishing mean oscillation, Trans. Amer. Math. Soc. 273 (1982), no. 2, 781–794. MR 667174, DOI 10.1090/S0002-9947-1982-0667174-2
- David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688, DOI 10.1090/S0273-0979-1981-14884-9
- Juha Kinnunen and Shulin Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations 24 (1999), no. 11-12, 2043–2068. MR 1720770, DOI 10.1080/03605309908821494
- Juha Kinnunen and Shulin Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients, Differential Integral Equations 14 (2001), no. 4, 475–492. MR 1799417
- Peter W. Jones, Extension theorems for BMO, Indiana Univ. Math. J. 29 (1980), no. 1, 41–66. MR 554817, DOI 10.1512/iumj.1980.29.29005
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Li He Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Math. Sin. (Engl. Ser.) 19 (2003), no. 2, 381–396. MR 1987802, DOI 10.1007/s10114-003-0264-4
Additional Information
- Sun-Sig Byun
- Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
- MR Author ID: 738383
- Email: byun@math.snu.ac.kr
- Lihe Wang
- Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242 – and – College of Sciences, Xian Jiaotong University, Xian 710049, People’s Republic of China
- Email: lwang@math.uiowa.edu
- Received by editor(s): August 5, 2005
- Published electronically: June 26, 2007
- Additional Notes: The first author was supported in part by KRF-2005-003-C00016.
The second author was supported in part by NSF Grant #0401261. - © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 5899-5913
- MSC (2000): Primary 35R05, 35R35; Secondary 35J15, 35J25
- DOI: https://doi.org/10.1090/S0002-9947-07-04238-9
- MathSciNet review: 2336309