Schauder estimates for a class of non-local elliptic equations

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388. doi: 10.1023/A:1016378210944.
[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.
[7]	L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society, Providence, RI, 1995.
[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.
[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.
[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.
[11]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001.
[12]	N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups," Imperial College Press, London, 2001. doi: 10.1142/9781860949746.
[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.
[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.
[15]	Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels, Manuscripta Math., 139 (2012), 291-319.
[16]	N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," American Mathematical Society, Providence, RI, 1996.
[17]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[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.
[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.
[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, ().
[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.
[22]	E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data, Ark. Mat., 19 (1981), 193-216. doi: 10.1007/BF02384477.
[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.
[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.
[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.

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.
[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.
[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.
[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.
[5]	R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388. doi: 10.1023/A:1016378210944.
[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.
[7]	L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society, Providence, RI, 1995.
[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.
[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.
[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.
[11]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001.
[12]	N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups," Imperial College Press, London, 2001. doi: 10.1142/9781860949746.
[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.
[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.
[15]	Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels, Manuscripta Math., 139 (2012), 291-319.
[16]	N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," American Mathematical Society, Providence, RI, 1996.
[17]	G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[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.
[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.
[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, ().
[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.
[22]	E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data, Ark. Mat., 19 (1981), 193-216. doi: 10.1007/BF02384477.
[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.
[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.
[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.

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