# American Institute of Mathematical Sciences

December  2015, 35(12): 5977-5998. doi: 10.3934/dcds.2015.35.5977

## Schauder estimates for solutions of linear parabolic integro-differential equations

 1 Department of Mathematics, The University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, United States 2 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

Received  April 2014 Revised  August 2014 Published  May 2015

We prove optimal pointwise Schauder estimates in the spatial variables for solutions of linear parabolic integro-differential equations. Optimal Hölder estimates in space-time for those spatial derivatives are also obtained.
Citation: Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
##### References:
 [1] B. Barrera, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators, and its application to nonlocal minimal surfaces,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XIII (2014), 609.  doi: 10.2422/2036-2145.201202_007.  Google Scholar [2] R. F. Bass, Regularity results for stable-like operators,, J. Funct. Anal., 257 (2009), 2693.  doi: 10.1016/j.jfa.2009.05.012.  Google Scholar [3] A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle,, Israel J. Math., 7 (1969), 254.  doi: 10.1007/BF02787619.  Google Scholar [4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math. (2), 130 (1989), 189.  doi: 10.2307/1971480.  Google Scholar [5] L. A. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar [6] L. A. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation,, Arch. Ration. Mech. Anal., 200 (2011), 59.  doi: 10.1007/s00205-010-0336-4.  Google Scholar [7] L. A. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations,, Ann. of Math. (2), 174 (2011), 1163.  doi: 10.4007/annals.2011.174.2.9.  Google Scholar [8] L. A. Caffarelli, C. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849.  doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar [9] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations,, Calc. Var. Partial Differential Equations, 49 (2014), 139.  doi: 10.1007/s00526-012-0576-2.  Google Scholar [10] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations II,, J. Differential Equations, 256 (2014), 130.  doi: 10.1016/j.jde.2013.08.016.  Google Scholar [11] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations,, Discrete Contin. Dyn. Syst., 33 (2013), 2319.  doi: 10.3934/dcds.2013.33.2319.  Google Scholar [12] H. Dong and S. Kim, Partial Schauder estimates for second-order elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 40 (2011), 481.  doi: 10.1007/s00526-010-0348-9.  Google Scholar [13] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Comm. Partial Differential Equations, 38 (2013), 1539.  doi: 10.1080/03605302.2013.808211.  Google Scholar [14] M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations,, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183.   Google Scholar [15] B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle,, Arch. Ration. Mech. Anal., 75 (1980), 51.  doi: 10.1007/BF00284620.  Google Scholar [16] D. Kriventsov, $C^{1,\alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels,, Comm. Partial Differential Equations, 38 (2013), 2081.  doi: 10.1080/03605302.2013.831990.  Google Scholar [17] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients,, Comm. Partial Differential Equations, 35 (2010), 1.  doi: 10.1080/03605300903424700.  Google Scholar [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translated from the Russian by S. Smith, (1968).   Google Scholar [19] Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composition material,, Comm. Pure Appl. Math., 56 (2003), 892.  doi: 10.1002/cpa.10079.  Google Scholar [20] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity,, Differential Integral Equations, 5 (1992), 1219.   Google Scholar [21] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time,, SIAM J. Math. Anal., 32 (2000), 588.  doi: 10.1137/S0036141098342842.  Google Scholar [22] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem,, Potential Anal., 40 (2014), 539.  doi: 10.1007/s11118-013-9359-4.  Google Scholar [23] J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels,, Calculus of Variations and Partial Differential Equations, (2014).  doi: 10.1007/s00526-014-0798-6.  Google Scholar [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).   Google Scholar [25] G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations,, Int. Math. Res. Not. IMRN, (2013), 3857.   Google Scholar

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##### References:
 [1] B. Barrera, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators, and its application to nonlocal minimal surfaces,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., XIII (2014), 609.  doi: 10.2422/2036-2145.201202_007.  Google Scholar [2] R. F. Bass, Regularity results for stable-like operators,, J. Funct. Anal., 257 (2009), 2693.  doi: 10.1016/j.jfa.2009.05.012.  Google Scholar [3] A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle,, Israel J. Math., 7 (1969), 254.  doi: 10.1007/BF02787619.  Google Scholar [4] L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations,, Ann. of Math. (2), 130 (1989), 189.  doi: 10.2307/1971480.  Google Scholar [5] L. A. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations,, Comm. Pure Appl. Math., 62 (2009), 597.  doi: 10.1002/cpa.20274.  Google Scholar [6] L. A. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation,, Arch. Ration. Mech. Anal., 200 (2011), 59.  doi: 10.1007/s00205-010-0336-4.  Google Scholar [7] L. A. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations,, Ann. of Math. (2), 174 (2011), 1163.  doi: 10.4007/annals.2011.174.2.9.  Google Scholar [8] L. A. Caffarelli, C. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849.  doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar [9] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations,, Calc. Var. Partial Differential Equations, 49 (2014), 139.  doi: 10.1007/s00526-012-0576-2.  Google Scholar [10] H. A. Chang Lara and G. Davila, Regularity for solutions of non local parabolic equations II,, J. Differential Equations, 256 (2014), 130.  doi: 10.1016/j.jde.2013.08.016.  Google Scholar [11] H. Dong and D. Kim, Schauder estimates for a class of non-local elliptic equations,, Discrete Contin. Dyn. Syst., 33 (2013), 2319.  doi: 10.3934/dcds.2013.33.2319.  Google Scholar [12] H. Dong and S. Kim, Partial Schauder estimates for second-order elliptic and parabolic equations,, Calc. Var. Partial Differential Equations, 40 (2011), 481.  doi: 10.1007/s00526-010-0348-9.  Google Scholar [13] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Comm. Partial Differential Equations, 38 (2013), 1539.  doi: 10.1080/03605302.2013.808211.  Google Scholar [14] M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations,, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183.   Google Scholar [15] B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle,, Arch. Ration. Mech. Anal., 75 (1980), 51.  doi: 10.1007/BF00284620.  Google Scholar [16] D. Kriventsov, $C^{1,\alpha}$ interior regularity for nonlinear nonlocal elliptic equations with rough kernels,, Comm. Partial Differential Equations, 38 (2013), 2081.  doi: 10.1080/03605302.2013.831990.  Google Scholar [17] N. V. Krylov and E. Priola, Elliptic and parabolic second-order PDEs with growing coefficients,, Comm. Partial Differential Equations, 35 (2010), 1.  doi: 10.1080/03605300903424700.  Google Scholar [18] O. A. Ladyzenskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type,, Translated from the Russian by S. Smith, (1968).   Google Scholar [19] Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composition material,, Comm. Pure Appl. Math., 56 (2003), 892.  doi: 10.1002/cpa.10079.  Google Scholar [20] G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity,, Differential Integral Equations, 5 (1992), 1219.   Google Scholar [21] L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time,, SIAM J. Math. Anal., 32 (2000), 588.  doi: 10.1137/S0036141098342842.  Google Scholar [22] R. Mikulevicius and H. Pragarauskas, On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem,, Potential Anal., 40 (2014), 539.  doi: 10.1007/s11118-013-9359-4.  Google Scholar [23] J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels,, Calculus of Variations and Partial Differential Equations, (2014).  doi: 10.1007/s00526-014-0798-6.  Google Scholar [24] E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).   Google Scholar [25] G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations,, Int. Math. Res. Not. IMRN, (2013), 3857.   Google Scholar
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