Advanced Search
Article Contents
Article Contents

Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains

Abstract Related Papers Cited by
  • In this paper we consider the global gradient estimates for weak solutions of $p(x)$-Laplacian type equation with small BMO coefficients in a $\delta$-Reifenberg flat domain. The modified Vitali covering lemma, good $\lambda$-inequalities, the maximal function technique and the appropriate localization method are the main analytical tools. The global Caldéron--Zygmund theory for such equations is obtained. Moreover, we generalize the regularity estimates in the Lebesgue spaces to the Orlicz spaces.
    Mathematics Subject Classification: Primary: 35R05, 35J92; Secondary: 35J15, 35J25.


    \begin{equation} \\ \end{equation}
  • [1]

    E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259.doi: 10.1007/s00205-002-0208-7.


    E. Acerbi and G. Mingione, Gradient estimates for the $p(x)$-Laplacean system, J. reine angew. Math., 584 (2005), 117-148.doi: 10.1515/crll.2005.2005.584.117.


    R. A. Adams and J. J. F. Fournier, Sobolev spaces, 2nd edition, Academic Press, New York, 2003.


    K. Astala, T. Iwaniec, P. Koskela and G. Martin, Mappings of BMO-bounded distortion, Math. Ann., 317 (2000), 703-726.doi: 10.1007/PL00004420.


    V. Bögelein and M. Parviainen, Self-improving property of nonlinear higher order parabolic systems near the boundary, NoDEA Nonlinear Differential Equations Appl., 17 (2010), 21-54.doi: 10.1007/s00030-009-0038-5.


    S. Byun and S. Ryu, Global weighted estimates for the gradient of solutions to nonlinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 291-313.doi: 10.1016/j.anihpc.2012.08.001.


    S. Byun and L. Wang, Elliptic equations with BMO coefficients in Reifenberg domains, Comm. Pure Appl. Math., 57 (2004), 1283-1310.doi: 10.1002/cpa.20037.


    S. Byun, L. Wang and S. Zhou, Nonlinear elliptic equations with small BMO coefficients in Reifenberg domains, J. Funct. Anal., 250 (2007), 167-196.doi: 10.1016/j.jfa.2007.04.021.


    S. Byun, F. Yao and S. Zhou, Gradient estimates in Orlicz space for nonlinear elliptic equations, J. Funct. Anal., 255 (2008), 1851-1873.doi: 10.1016/j.jfa.2008.09.007.


    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.doi: 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.3.CO;2-N.


    Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66 (2006), 1383-1406.doi: 10.1137/050624522.


    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), 1107-1134.doi: 10.2307/2375066.


    L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, vol. 2017, Springer-Verlag, Berlin, 2011.doi: 10.1007/978-3-642-18363-8.


    X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl., 263 (2001), 424-446.doi: 10.1006/jmaa.2000.7617.


    F. Giannetti and A. Passarelli di Napoli, Regularity results for a new class of functionals with non-standard growth conditions, J. Differential Equations, 254 (2013), 1280-1305.doi: 10.1016/j.jde.2012.10.011.


    P. Harjulehto, Variable exponent Sobolev spaces with zero boundary values, Math. Bohem., 132 (2007), 125-136.


    T. Iwaniec, Projections onto gradient fields and $L^p$-estimates for degenerated elliptic operators, Studia Math., 75 (1983), 293-312.


    T. Iwaniec, $p$-harmonic tensors and quasiregular mappings, Ann. Math., 136 (1992), 589-624.doi: 10.2307/2946602.


    T. Iwaniec and A. Verde, On the operator $\mathcalL(f)=f\log|f|$, J. Funct. Anal., 169 (1999), 391-420.doi: 10.1006/jfan.1999.3443.


    J. Kinnumen and S. Zhou, A local estimate for nonlinear equations with discontinuous coefficients, Comm. Partial Differential Equations, 24 (1999), 2043-2068.doi: 10.1080/03605309908821494.


    J. Kinnunen and S. Zhou, A boundary estimate for nonlinear equations with discontinuous coefficients, Differential Integral Equations, 14 (2001), 475-492.


    V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Scientific, 1991.doi: 10.1142/9789814360302.


    T. Kilpeläinen and P. Koskela, Global integrability of the gradients of solutions to partial differential equations, Nonlinear Anal., 23 (1994), 899-909.doi: 10.1016/0362-546X(94)90127-9.


    O. Kováčik and J. Rákosník, On spaces $L^{p(x)}$ and $W^{k,p(x)}$, Czechoslovak Math. J., 41 (1991), 592-618.


    G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.doi: 10.1016/0362-546X(88)90053-3.


    T. Mengesha and N. C. Phuc, Global estimates for quasilinear elliptic equations on Reifenberg flat domains, Arch. Ration. Mech. Anal., 203 (2012), 189-216.doi: 10.1007/s00205-011-0446-7.


    K. Rajagopal and M. Růžička, Mathematical modeling of electro-rheological fluids, Contin. Mech. Thermodyn., 13 (2001), 59-78.


    M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, in: Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.doi: 10.1007/BFb0104029.


    J. Musielak, Orlicz Spaces and Modular Spaces, Springer-Verlag, Berlin, 1983.


    M. Rao and Z. Ren, Applications of Orlicz Spaces, Marcel Dekker Inc., New York, 2002.doi: 10.1201/9780203910863.


    E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993.


    L. Wang, F. Yao, S. Zhou and H. Jia, Optimal regularity for the poisson equation, Proc. Amer. Math. Soc., 137 (2009), 2037-2047.doi: 10.1090/S0002-9939-09-09805-0.


    V. V. Zhikov, On some variational problems, Russ. J. Math. Phys., 5 (1997), 105-116.

  • 加载中

Article Metrics

HTML views() PDF downloads(218) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint