Skip to main content
SpringerLink
Log in

Global Estimates for Quasilinear Elliptic Equations on Reifenberg Flat Domains

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

New global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems. The coefficients are assumed to have small BMO seminorms, and the boundary of the domain is sufficiently flat in the sense of Reifenberg. The main regularity estimates obtained are in weighted Lorentz spaces. Other regularity results in Lorentz–Morrey, Morrey, and Hölder spaces are shown to follow from the main estimates.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams D.R., Lewis J.L.: On Morrey-Besov inequalities. Stud. Math. 74, 169–182 (1982)

    MATH  MathSciNet  Google Scholar 

  3. Auscher P., Qafsaoui M.: Observations on W 1, p estimates for divergence elliptic equations with VMO coefficients. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 5(8), 487–509 (2002)

    MATH  MathSciNet  Google Scholar 

  4. Byun S., Wang L.: Elliptic equations with BMO nonlinearity in Reifenberg domains. Adv. Math. 219, 1937–1971 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Byun S., Wang L.: Elliptic equations with BMO coefficients in Reifenberg domains. Commun. Pure Appl. Math. 57, 1283–1310 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  6. Byun S., Wang L., Zhou S.: Nonlinear elliptic equations with BMO coefficients in Reifenberg domains. J. Funct. Anal. 250, 167–196 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Byun S., Yao F., Zhou S.: Gradient estimates in Orlicz space for nonlinear elliptic equations. J. Funct. Anal. 250, 1851–1873 (2008)

    Article  MathSciNet  Google Scholar 

  8. Caffarelli L., Peral I.: On W 1,  p estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51, 1–21 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chung H.-M., Hunt R.A., Kurtz D.: The Hardy-Littlewood maximal function on L(p, q) spaces with weights. Indiana U. Math. J. 31, 109–120 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  10. Coifman R.R., Fefferman C.: Weighted norm inequalities for maximal functions and singular integrals. Stud. Math. 51, 241–250 (1974)

    MATH  MathSciNet  Google Scholar 

  11. DiBenedetto E., Manfredi J.: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. Am. J. Math. 115, 1107–1134 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  12. Di Fazio G.: L p estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Unione Mat. Ital. A (7) 10, 409–420 (1996)

    MATH  MathSciNet  Google Scholar 

  13. Evans, C.L.: Partial differential equations. Graduate Studies in Mathematics, Vol. 19. American Mathematical Society, Providence, RI, 1998, xviii+662 pp

  14. Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (1983)

    MATH  Google Scholar 

  15. Grafakos L.: Classical and Modern Fourier Analysis. Pearson Education, Inc., Upper Saddle River (2004)

    MATH  Google Scholar 

  16. Guadalupe J., Perez M.: Perturbation of orthogonal Fourier expansions. J. Approx. Theory 92, 294–307 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hajlasz P., Martio O.: Traces of Sobolev functions on fractal type sets and characterization of extension domains. J. Funct. Anal. 143, 221–246 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hunt R.A., Kurtz D.: The Hardy-Littlewood maximal function on L(p, 1). Indiana U. Math. J. 32, 155–158 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  19. Iwaniec T.: Projections onto gradient fields and L p-estimates for degenerated elliptic operators. Stud. Math. 75, 293–312 (1983)

    MATH  MathSciNet  Google Scholar 

  20. Iwaniec T., Koskela P., Martin G.: Gaven mappings of BMO-distortion and Beltrami-type operators. J. Anal. Math. 88, 337–381 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Jerison D., Kenig C.: The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jones P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  23. Kenig C., Toro T.: Free boundary regularity for harmonic measures and the Poisson kernel. Ann. Math. 150, 367–454 (1999)

    Article  MathSciNet  Google Scholar 

  24. Kenig C., Toro T.: Poisson kernel characterization of Reifenberg flat chord arc domains. Ann. Sci. Ecole Norm. Sup. (4) 36, 323–401 (2003)

    MATH  MathSciNet  Google Scholar 

  25. Kinnunen J., Zhou S.: A local estimate for nonlinear equations with discontinuous coefficients. Commun. Partial Differ. Equ. 24, 2043–2068 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kinnunen J., Zhou S.: A boundary estimate for nonlinear equations with discontinuous coefficients. Differ. Integr. Equ. 14, 475–492 (2001)

    MATH  MathSciNet  Google Scholar 

  27. Lieberman G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  28. Meyers N.G.: An L p-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa (3) 17, 189–206 (1963)

    MATH  MathSciNet  Google Scholar 

  29. Mengesha T., Phuc N.C.: Weighted and regularity estimates for nonlinear equations on Reifenberg flat domains. J. Differ. Equ. 250(1), 2485–2507 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  30. Milman, M.: Rearrangements of BMO functions and interpolation. Lecture Notes in Mathematics, Vol. 1070. Springer, Berlin, 1984

  31. Mingione G.: Gradient estimates below the duality exponent. Math. Ann. 346, 571–627 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  32. Mingione G.: Towards a non-linear Calderon-Zygmund theory. Quaderni di Matematica 23, 371–458 (2009)

    MathSciNet  Google Scholar 

  33. Muckenhoupt B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  34. Phuc N.C.: Quasilinear Riccati type equations with super-critical exponents. Commun. Partial Differ. Equ. 35, 1958–1981 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Phuc N.C.: Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 10, 1–17 (2011)

    MATH  MathSciNet  Google Scholar 

  36. Ragusa M.A.: Regularity of solutions of divergence form elliptic equations. Proc. Am. Math. Soc. 128, 533–540 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  37. Reinfenberg E.: Solutions of the Plateau problem for m-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)

    Article  MathSciNet  Google Scholar 

  38. Sarason D.: Functions of vanishing mean oscillation. Trans. Am. Math. Soc. 207, 391–405 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  39. Semmes S.: Hypersurfaces in \({\mathbb{R}^n}\) whose unit normal has small BMO norm. Proc. Am. Math. Soc. 112, 403–412 (1991)

    MATH  MathSciNet  Google Scholar 

  40. Toro T.: Doubling and flatness: geometry of measures. Not. Am. Math. Soc. 44, 1087–1094 (1997)

    MATH  MathSciNet  Google Scholar 

  41. Vuorinen M., Martio O., Ryazanov V.: On the local behavior of quasiregular mappings in n-dimensional space. Izv. Math. 62, 1207–1220 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang L.: A geometric approach to the Calderoń-Zygmund estimates. Acta Math. Sin. (Engl. Ser.) 19, 381–396 (2003)

    Article  MathSciNet  Google Scholar 

  43. Widman K.O.: Inequalities for the Green functions and boundary continuity of the gradient of solutions of elliptic differential equations. Math. Scand. 21, 17–37 (1967)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, 303 Lockett Hall, Baton Rouge, LA, 70803, USA

    Tadele Mengesha & Nguyen Cong Phuc

Authors
  1. Tadele Mengesha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Cong Phuc
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tadele Mengesha.

Additional information

Communicated by G. Dal Maso

Tadele Mengesha’s research was done while the author was on leave of absence from Coastal Carolina University (CCU). The author acknowledges the support of CCU. The research is also supported by NSF grant DMS-0406374. Nguyen Cong Phuc was supported in part by NSF grant DMS-0901083.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mengesha, T., Phuc, N.C. Global Estimates for Quasilinear Elliptic Equations on Reifenberg Flat Domains. Arch Rational Mech Anal 203, 189–216 (2012). https://doi.org/10.1007/s00205-011-0446-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-011-0446-7

Keywords

Search

Navigation