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Quasilinear elliptic equations with BMO coefficients in Lipschitz domains


Authors: Sun-Sig Byun and Lihe Wang
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
Published electronically: June 26, 2007
MathSciNet review: 2336309
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Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. S. Byun, Elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc., 357 (2005), 1025-1046. MR 2110431 (2005i:35054)
  • 2. L.A. Caffarelli and X. Cabré, Fully nonlinear elliptic equations, vol. 43, Amer. Math. Soc., Providence, RI, 1995. MR 1351007 (96h:35046)
  • 3. L.A. Caffarelli and I. Peral, On $ W^{1,p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math., 51, (1998), 1-21.MR 1486629 (99c:35053)
  • 4. E. DiBenedetto and J. Manfredi, On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems, Amer. J. Math., 115 (5), 1993, 1107-1134.MR 1246185 (94i:35077)
  • 5. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, (1998), xviii+662 pp. ISBN: 0-8218-0772-2.MR 1625845 (99e:35001)
  • 6. T. Iwaniec, Projections onto gradient fields and $ L^p$-estimates for degenerate elliptic operators, Studia Math., 75, 1983, 293-312.MR 0722254 (85i:46037)
  • 7. D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130, (1995), 161-219. MR 1331981 (96b:35042)
  • 8. D. Jerison and C. Kenig, The logarithm of the Poisson kernel of a $ C^{1}$ domain has vanishing mean oscillation, Trans. Amer. Math. Soc. 273, (1982), 781-794.MR 0667174 (83k:31004)
  • 9. D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.), 4, (1981), 203-207. MR 0598688 (84a:35064)
  • 10. J. Kinnunen and S. Zhou, A Local estimate for nonlinear equations with discontinuous coefficients, Comm. partial differential equations, 24, (1999), 2043-2068. MR 1720770 (2000k:35084)
  • 11. J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients, Differential and integral equations, 14 (2001), 475-492. MR 1799417 (2002a:35070)
  • 12. P.W. Jones, Extension theorems for BMO, Indiana Univ. Math. J., 29, (1980), 41-66.MR 0554817 (81b:42047)
  • 13. E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • 14. L. Wang, A geometric approach to the Calderón-Zygmund estimates, Acta Mathematica Sinica, 19, (2003), 381-396. MR 1987802 (2004e:42033)

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Additional Information

Sun-Sig Byun
Affiliation: Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
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

DOI: https://doi.org/10.1090/S0002-9947-07-04238-9
Keywords: $W^{1,p}$ estimates, quasilinear elliptic equations, BMO space, Lipschitz domain, maximal function, Vitali covering lemma
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.
Article copyright: © Copyright 2007 American Mathematical Society

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