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June  2013, 33(6): 2319-2347. doi: 10.3934/dcds.2013.33.2319

Schauder estimates for a class of non-local elliptic equations

Hongjie Dong 1, and  Doyoon Kim 2,

1. 

Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912
2. 

Department of Applied Mathematics, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin-si, Gyeonggi-do 446-701, South Korea

Received  January 2012 Revised  May 2012 Published  December 2012

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.
Keywords: Lévy processes., Non-local elliptic equations, Schauder estimates.
Mathematics Subject Classification: Primary: 45K05, 35B65; Secondary: 60J7.
Citation: Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2319-2347. doi: 10.3934/dcds.2013.33.2319
References:
[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.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the corresponding martingale problem,, preprint, ().   Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

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

[24]

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.  Google Scholar

[25]

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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

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

[3]

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.  Google Scholar

[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.  Google Scholar

[5]

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

[6]

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.  Google Scholar

[7]

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

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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.  Google Scholar

[19]

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.  Google Scholar

[20]

R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the corresponding martingale problem,, preprint, ().   Google Scholar

[21]

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.  Google Scholar

[22]

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

[23]

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

[24]

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.  Google Scholar

[25]

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.  Google Scholar

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