Advanced Search
Article Contents
Article Contents

Schauder estimates for a class of non-local elliptic equations

Abstract Related Papers Cited by
  • We prove Schauder estimates for a class of non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$ and either Dini or Hölder continuous data. Here $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function, which is not necessarily to be homogeneous, regular, or symmetric. As an application, we prove that the operators give isomorphisms between the Lipschitz--Zygmund spaces $\Lambda^{\alpha+\sigma}$ and $\Lambda^\alpha$ for any $\alpha>0$. Several local estimates and an extension to operators with kernels $K(x,y)$ are also discussed.
    Mathematics Subject Classification: Primary: 45K05, 35B65; Secondary: 60J75.


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

    H. Abels and M. Kassmann, The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels, Osaka J. Math., 46 (2009), 661-683.


    R. F. Bass, Regularity results for stable-like operators, J. Funct. Anal., 257 (2009), 2693-2722.doi: 10.1016/j.jfa.2009.05.012.


    R. F. Bass and M. Kassmann, Harnack inequalities for non-local operators of variable order, Trans. Amer. Math. Soc., 357 (2005), 837-850.doi: 10.1090/S0002-9947-04-03549-4.


    R. F. Bass and M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order, Comm. Partial Differential Equations, 30 (2005), 1249-1259.doi: 10.1080/03605300500257677.


    R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388.doi: 10.1023/A:1016378210944.


    G. Barles, E. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations, J. Eur. Math. Soc., 13 (2011), 1-26.doi: 10.4171/JEMS/242.


    L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society, Providence, RI, 1995.


    L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math., 62 (2009), 597-638.doi: 10.1002/cpa.20274.


    L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Ration. Mech. Anal., 200 (2011), 59-88.doi: 10.1007/s00205-010-0336-4.


    H. Dong and D. Kim, On $L_p$-estimates for a class of non-local elliptic equations, J. Funct. Anal., 262 (2012), 1166-1199.doi: 10.1016/j.jfa.2011.11.002.


    D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001.


    N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups," Imperial College Press, London, 2001.doi: 10.1142/9781860949746.


    N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. II. Generators and Their Potential Theory," Imperial College Press, London, 2001.doi: 10.1142/9781860949562.


    M. Kassmann, A priori estimates for integro-differential operators with measurable kernels, Calc. Var. Partial Differential Equations, 34 (2009), 1-21.doi: 10.1007/s00526-008-0173-6.


    Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels, Manuscripta Math., 139 (2012), 291-319.


    N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," American Mathematical Society, Providence, RI, 1996.


    G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.


    R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Liet. Mat. Rink., 32 (1992), 299-331; translation in Lithuanian Math. J., 32 (1992), 238-264.doi: 10.1007/BF02450422.


    R. Mikulevicius and H. Pragarauskas, On Hölder solutions of the integro-differential Zakai equation, Stochastic Process. Appl., 119 (2009), 3319-3355.doi: 10.1016/j.spa.2009.05.008.


    R. Mikulevicius and H. PragarauskasOn the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the corresponding martingale problem, preprint, arXiv:1103.3492.


    L. Silvestre, Hölder estimates for solutions of integro-differential equations like the fractional laplace, Indiana Univ. Math. J., 55 (2006), 1155-1174.doi: 10.1512/iumj.2006.55.2706.


    E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data, Ark. Mat., 19 (1981), 193-216.doi: 10.1007/BF02384477.


    E. M. Stein, "Singular Integrals and Differentiability Properties of Functions," Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J., 1970, xiv+290.


    E. M. Stein, "Harmonic Analysis: Real-Variable Methods Orthogonality, and Oscillatory Integrals," Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695.


    X.-J. Wang, Schauder estimates for elliptic and parabolic equations, Chinese Ann. Math. Ser. B, 27 (2006), 637-642.doi: 10.1007/s11401-006-0142-3.

  • 加载中

Article Metrics

HTML views() PDF downloads(199) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint