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Schauder estimates for a class of non-local elliptic equations
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 |
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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