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 and Continuous Dynamical Systems, 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), 609-639. doi: 10.2422/2036-2145.201202_007.

[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]

A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle, Israel J. Math., 7 (1969), 254-262. doi: 10.1007/BF02787619.

[4]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130 (1989), 189-213. 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), 597-638. 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), 59-88. 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), 1163-1187. 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), 849-869. 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), 139-172. 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), 130-156. 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), 2319-2347. 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), 481-500. 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), 1539-1573. 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), 183-212.

[15]

B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Ration. Mech. Anal., 75 (1980), 51-58. 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), 2081-2106. 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), 1-22. 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, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1968.

[19]

Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composition material, Comm. Pure Appl. Math., 56 (2003), 892-925. 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), 1219-1236.

[21]

L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. 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), 539-563. 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, arXiv:1401.4521, 2014. doi: 10.1007/s00526-014-0798-6.

[24]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.

[25]

G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Not. IMRN, (2013), 3857-3877.

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), 609-639. doi: 10.2422/2036-2145.201202_007.

[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]

A. Brandt, Interior Schauder estimates for parabolic differential- (or difference-) equations via the maximum principle, Israel J. Math., 7 (1969), 254-262. doi: 10.1007/BF02787619.

[4]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math. (2), 130 (1989), 189-213. 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), 597-638. 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), 59-88. 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), 1163-1187. 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), 849-869. 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), 139-172. 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), 130-156. 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), 2319-2347. 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), 481-500. 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), 1539-1573. 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), 183-212.

[15]

B. F. Knerr, Parabolic interior Schauder estimates by the maximum principle, Arch. Ration. Mech. Anal., 75 (1980), 51-58. 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), 2081-2106. 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), 1-22. 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, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I. 1968.

[19]

Y. Y. Li and L. Nirenberg, Estimates for elliptic system from composition material, Comm. Pure Appl. Math., 56 (2003), 892-925. 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), 1219-1236.

[21]

L. Lorenzi, Optimal Schauder estimates for parabolic problems with data measurable with respect to time, SIAM J. Math. Anal., 32 (2000), 588-615. 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), 539-563. 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, arXiv:1401.4521, 2014. doi: 10.1007/s00526-014-0798-6.

[24]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J. 1970.

[25]

G. Tian and X.-J.Wang, A priori estimates for fully nonlinear parabolic equations, Int. Math. Res. Not. IMRN, (2013), 3857-3877.

[1]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

[2]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051

[3]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025

[4]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17

[5]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053

[6]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160

[7]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053

[8]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191

[9]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541

[10]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065

[11]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems and Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693

[12]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015

[13]

Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial and Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119

[14]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044

[15]

Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059

[16]

Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249

[17]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677

[18]

Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026

[19]

Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete and Continuous Dynamical Systems, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069

[20]

Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (172)
  • HTML views (0)
  • Cited by (21)

Other articles
by authors

[Back to Top]