Skip to main content
SpringerLink
Log in

\(W^{2,p(\cdot )}\)-regularity for elliptic equations in nondivergence form with BMO coefficients

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We establish an optimal \(W^{2,p(\cdot )}\)-estimate to the Dirichlet problem for an elliptic equation in nondivergence form with discontinuous coefficients on a \(C^{1,1}\) bounded domain for every variable exponent \(p(\cdot )\) with log-Hölder continuity. The matrix of the coefficients is assumed to have a small BMO semi-norm, depending on the exponent, the boundary of the domain, and the matrix itself.

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. Acerbi, E., Mingione, G.: Regularity results for a class of functionals with non-standard growth. Arch. Ration. Mech. Anal. 156, 121–140 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Acerbi, E., Mingione, G.: Gradient estimates for the \(p(x)\)-Laplacean system. J. Reine Angew. Math. 584, 117–148 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  3. Acerbi, E., Mingione, G.: Gradient estimates for a class of parabolic systems. Duke Math. J. 136, 285–320 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Baroni, P., Bögelein, V.: Calderón-Zygmund estimates for parabolic \(p(x, t)\)-Laplacian systems. Rev. Mat. Iberoam. 30(4), 1355–1386 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bögelein, V., Duzaar, F.: Hölder estimates for parabolic \(p(x, t)\)-Laplacian systems. Math. Ann. 354(3), 907–938 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Byun, S., Ok, J., Ryu, S.: Global gradient estimates for elliptic equations of \(p(x)\)-Laplacian type with BMO nonlinearity. J. Reine Angew. Math. (2015). doi:10.1515/crelle-2014-0004

  7. Byun, S., Ok, J., Wang, L.: \(W^{1, p(\cdot )}\)-regularity for elliptic equations with measurable coefficients in nonsmooth domains. Commun. Math. Phys. 329(3), 937–958 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006) (electronic)

  9. Chiarenza, F., Frasca, M., Longo, P.: Interior \(W^{2, p}\) estimates for nondivergence elliptic equations with discontinuous coefficients. Ricerche Mat. 40(1), 149–168 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Chiarenza, F., Frasca, M., Longo, P.: \(W^{2, p}\)-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients. Trans. Am. Math. Soc. 336(2), 841–853 (1993)

    MATH  MathSciNet  Google Scholar 

  11. Diening, L.: Maximal function on generalized Lebesgue spaces \(L^{p(\cdot )}\). Math. Inequal. Appl. 7(2), 245–253 (2004)

    MATH  MathSciNet  Google Scholar 

  12. Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics, vol. 2017. Springer, Heidelberg (2011)

  13. Diening, L., Lengeler, D., Růžička, M.: The Stokes and Poisson problem in variable exponent spaces. Complex Var. Elliptic Equ. 56, 789–811 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Diening, L., Růžička, M.: Calderón–Zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot )}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Fan, X.: Global \(C^{1,\alpha }\) regularity for variable exponent elliptic equations in divergence form. J. Differ. Equ. 235, 397–417 (2007)

    Article  MATH  Google Scholar 

  16. Franciosi, M., Fusco, N.: \(W^{2, p}\) regularity for the solutions of elliptic nondivergence form equations with rough coefficients. Ricerche Mat. 38(1), 93–106 (1989)

    MATH  MathSciNet  Google Scholar 

  17. Fu, Y., Zang, A.: Interpolation inequalities for derivatives in variable exponent Lebesgue-Sobolev spaces. Nonlinear Anal. 69(10), 3629–3636 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften, vol. 224. Springer, Berlin (1977)

    Book  Google Scholar 

  19. Han, Q., Lin, F.H.: Elliptic Partial Differential Equations, Courant Lecture Notes in Mathematics, vol. 1. New York University, Courant Institute of Mathematical Sciences, New York, American Mathematical Society, Providence (1997)

  20. Kim, D., Krylov, N.V.: Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others. SIAM J. Math. Anal. 39(2), 489–506 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41(116), 592–618 (1991)

  22. Mingione, G.: The Calderón-Zygmund theory for elliptic problems with measure data. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6, 195–261 (2007)

  23. Maugeri, A., Palagachev, D.K., Softova, L.G.: Elliptic and Parabolic Equations with Discontinuous Coefficients, Mathematical Research, vol. 109. Wiley-VCH Verlag Berlin GmbH, Berlin (2000)

    Book  Google Scholar 

  24. Rajagopal, K.R., Růžička, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)

    Article  MATH  Google Scholar 

  25. Růžička, M.: Electrorheological Fluids: Modeling and mathematical theory, Springer Lecture Notes in Math, vol. 1748. Springer, Berlin (2000)

    Google Scholar 

  26. Samko, S.G.: Density of \(C^{\infty }_0(\mathbb{R}^n)\) in the generalized Sobolev spaces \(W^{m, p(x)}(\mathbb{R}^n)\) (Russian). Dokl. Akad. Nauk. 369(4), 451–454 (1999)

    MathSciNet  Google Scholar 

  27. Samko, S.G.: Convolution and potential type operators in \(L^{p(x)}(\mathbb{R}^n)\). Integral Transform. Spec. Funct. 7(3–4), 261–284 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  28. Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cords coefficients. SIAM J. Numer. Anal. 51(4), 2088–2106 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  29. Smears, I., Süli, E.: Discontinuous Galerkin finite element approximation of Hamilton-Jacobi-Bellman equations with Cordes coefficients. SIAM J. Numer. Anal. 52(2), 993–1016 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  30. Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 233. Springer, Berlin (1979)

    Google Scholar 

  31. Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50(4), 675–710, 877 (1986)

  32. Zhikov, V.V.: Meyer-type estimates for solving the nonlinear Stokes system. Differ. Equ. 33(1), 108–115 (1997)

    MATH  MathSciNet  Google Scholar 

  33. Zhikov, V.V., Kozlov, S.M., Ole\(\check{\rm i}\)nik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, New York (1994)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences and Research Institute of Mathematics, Seoul National University, Seoul, 151-747, Korea

    Sun-Sig Byun

  2. Department of Mathematical Sciences, Seoul National University, Seoul, 151-747, Korea

    Mikyoung Lee

  3. School of Mathematics, Korea Institute for Advanced Study, Seoul, 130-722, Korea

    Jihoon Ok

Authors
  1. Sun-Sig Byun
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikyoung Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jihoon Ok
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikyoung Lee.

Additional information

S. Byun was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2009-2012R1A2A2A01047030). J. Ok was supported by TJ Park Science Fellowship of POSCO TJ Park Foundation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Byun, SS., Lee, M. & Ok, J. \(W^{2,p(\cdot )}\)-regularity for elliptic equations in nondivergence form with BMO coefficients. Math. Ann. 363, 1023–1052 (2015). https://doi.org/10.1007/s00208-015-1194-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-015-1194-z

Mathematics Subject Classification

Search

Navigation