Skip to main content
SpringerLink
Log in

Nonlinear gradient estimates for elliptic equations in quasiconvex domains

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We study a general model of nonvariational elliptic equations of \(p\)-Laplacian type in quasiconvex domains, which are locally approximated by convex domains. We prove that both the gradient and the associated nonhomogeneous term belong to the same \(L^q\) space for every \(q \in [p, \infty )\). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction. In addition, we extend our result in Lebesgue spaces to Orlicz spaces.

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. Adolfsson, V.: \(L^p\)-integrability of the second order derivatives of Green potentials in convex domains. Pacific J Math 159(2), 201–225 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  2. Byun, S.: Gradient estimates in Orlicz spaces for nonlinear elliptic equations with BMO nonlinearity in nonsmooth domains. Forum Math 23(4), 693–711 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Byun, S., Wang, L.: Elliptic equations with BMO coefficients in Reifenberg domains. Comm Pure Appl Math 57(10), 1283–1310 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Byun, S., Wang, L.: Elliptic equations with BMO nonlinearity in Reifenberg domains. Adv Math 219(6), 1937–1971 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Byun, S., Wang, L.: Nonlinear gradient estimates for elliptic equations of general type. Calc Var Partial Differ Equ 45(3–4), 403–419 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Byun, S., Yao, P., Zhou, S.: Gradient estimates in Orlicz space for nonlinear elliptic equations. J Funct Anal 255(8), 1851–1873 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caffarelli, L.A., Peral, I.: On \(W^{1, p}\) estimates for elliptic equations in divergence form. Comm Pure Appl Math 51(1), 1–21 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cianchi, A., Maz’ya, V.G.: Global Lipschitz regularity for a class of quasilinear elliptic equations. Comm Partial Differ Equ 36(1), 100–133 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cianchi, A., Maz’ya, V.G.: Gradient regularity via rearrangements for p -Laplacian type elliptic boundary value problems. J Eur Math Soc 16(3), 571–595 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cianchi, A., Maz’ya, V.G.: Global boundedness of the gradient for a class of nonlinear elliptic systems. Arch Ration Mech Anal 212(1), 129–177 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. David, G., Toro, T.: Reifenberg flat metric spaces, snowballs, and embeddings. Math Ann 315(4), 641–710 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. Palagachev, D.K., Softova, L.G.: Quasilinear divergence form parabolic equations in Reifenberg flat domains. Discrete Contin Dyn Syst 31(4), 1397–1410 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Di Fazio, G.: \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients. Boll Un Mat Ital 10(2), 409–420 (1996)

    MathSciNet  MATH  Google Scholar 

  14. Hajłasz, P., Martio, O.: Traces of Sobolev functions on fractal type sets and characterization of extension domains. J Funct Anal 143(1), 221–246 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Leray, J., Lions, J.L.: Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder. Bull Soc Math France 93, 97–107 (1965)

    MathSciNet  MATH  Google Scholar 

  16. Jia, H., Li, D., Wang, L.: Global regularity for divergence form elliptic equations on quasiconvex domains. J Differ Equ 249(12), 3132–3147 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jia, H., Li, D., Wang, L.: Global regularity for divergence form elliptic equations in Orlicz spaces on quasiconvex domains. Nonlinear Anal 74(4), 1336–1344 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jia, H., Wang, L.: Divergence form parabolic equations on time-dependent quasiconvex domains. Int J Math 23(12), 17 (2012)

    Article  MathSciNet  Google Scholar 

  19. Jozef, K.: On boundedness of the weak solution for some class of quasilinear partial differential equations. Časopis Pěst Mat 98, 43–55 (1973)

    MATH  Google Scholar 

  20. Kokilashvili, V., Krbec, M.: Weighted inequalities in Lorentz and Orlicz spaces. World Scientific Publishing Co., River Edge (1991)

    Book  MATH  Google Scholar 

  21. Mingione, G.: Calderón-Zygmund estimates for measure data problems. C R Math Acad Sci Paris 344(7), 437–442 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Phuc, N.C.: Weighted estimates for nonhomogeneous quasilinear equations with discontinuous coefficients. Ann Sc Norm Super Pisa Cl Sci 10(1), 1–17 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Marcel Dekker, New York (1991)

    MATH  Google Scholar 

  24. Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  25. Wang, L., Yao, F., Zhou, S., Jia, H.: Optimal regularity for the Poisson equation. Proc Amer Math Soc 137(6), 2037–2047 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yao, F.: Higher-integrability for a quasilinear parabolic equation of p -Laplacian type. Nonlinear Anal 70(3), 1265–1274 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yao, F., Sun, Y., Zhou, S.: Gradient estimates in Orlicz spaces for quasilinear elliptic equation. Nonlinear Anal 69(8), 2553–2565 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Science, Seoul National University, Seoul, 151-747, Republic of Korea

    Sun-Sig Byun, Hun Kwon & Hyoungsuk So

  2. Department of Mathematics, University of Iowa City, Iowa City, IA, 52242, USA

    Lihe Wang

Authors
  1. Sun-Sig Byun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hun Kwon
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Hyoungsuk So
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lihe Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hyoungsuk So.

Additional information

Communicated by L. Caffarelli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Byun, SS., Kwon, H., So, H. et al. Nonlinear gradient estimates for elliptic equations in quasiconvex domains. Calc. Var. 54, 1425–1453 (2015). https://doi.org/10.1007/s00526-015-0830-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-015-0830-5

Mathematics Subject Classification

Search

Navigation