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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations,, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183.

[15]

B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle,, Arch. Ration. Mech. Anal., 75 (1980), 51. doi: 10.1007/BF00284620.

[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.

[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.

[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).

[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.

[20]

G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity,, Differential Integral Equations, 5 (1992), 1219.

[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.

[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.

[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.

[24]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[25]

G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations,, Int. Math. Res. Not. IMRN, (2013), 3857.

show all references

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations,, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183.

[15]

B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle,, Arch. Ration. Mech. Anal., 75 (1980), 51. doi: 10.1007/BF00284620.

[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.

[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.

[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).

[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.

[20]

G. M. Lieberman, Intermediate Schauder theory for second order parabolic equations. IV. Time irregularity and regularity,, Differential Integral Equations, 5 (1992), 1219.

[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.

[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.

[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.

[24]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970).

[25]

G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations,, Int. Math. Res. Not. IMRN, (2013), 3857.

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