June  2013, 33(6): 2319-2347. doi: 10.3934/dcds.2013.33.2319

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

Received  January 2012 Revised  May 2012 Published  December 2012

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.
Citation: Hongjie Dong, Doyoon Kim. Schauder estimates for a class of non-local elliptic equations. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388. doi: 10.1023/A:1016378210944.  Google Scholar

[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.  Google Scholar

[7]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society, Providence, RI, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups," Imperial College Press, London, 2001. doi: 10.1142/9781860949746.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels, Manuscripta Math., 139 (2012), 291-319. Google Scholar

[16]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," American Mathematical Society, Providence, RI, 1996.  Google Scholar

[17]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[22]

E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data, Ark. Mat., 19 (1981), 193-216. doi: 10.1007/BF02384477.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

R. F. Bass and D. A. Levin, Harnack inequalities for jump processes, Potential Anal., 17 (2002), 375-388. doi: 10.1023/A:1016378210944.  Google Scholar

[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.  Google Scholar

[7]

L. Caffarelli and X. Cabré, "Fully Nonlinear Elliptic Equations," American Mathematical Society, Providence, RI, 1995.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 ed. Springer-Verlag, Berlin, 2001.  Google Scholar

[12]

N. Jacob, "Pseudo Differential Operators and Markov Processes. Vol. I. Fourier Analysis and Semigroups," Imperial College Press, London, 2001. doi: 10.1142/9781860949746.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

Y. Kim and K. Lee, Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels, Manuscripta Math., 139 (2012), 291-319. Google Scholar

[16]

N. V. Krylov, "Lectures on Elliptic and Parabolic Equations in Hölder Spaces," American Mathematical Society, Providence, RI, 1996.  Google Scholar

[17]

G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[22]

E. Sperner, Schauder's existence theorem for $\alpha$-Dini continuous data, Ark. Mat., 19 (1981), 193-216. doi: 10.1007/BF02384477.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Yuanhong Wei, Xifeng Su. On a class of non-local elliptic equations with asymptotically linear term. Discrete & Continuous Dynamical Systems, 2018, 38 (12) : 6287-6304. doi: 10.3934/dcds.2018154

[2]

Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010

[3]

Huxiao Luo, Xianhua Tang, Zu Gao. Sign-changing solutions for non-local elliptic equations with asymptotically linear term. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1147-1159. doi: 10.3934/cpaa.2018055

[4]

Raffaella Servadei, Enrico Valdinoci. Variational methods for non-local operators of elliptic type. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 2105-2137. doi: 10.3934/dcds.2013.33.2105

[5]

Wei Zhong, Yongxia Zhao, Ping Chen. Equilibrium periodic dividend strategies with non-exponential discounting for spectrally positive Lévy processes. Journal of Industrial & Management Optimization, 2021, 17 (5) : 2639-2667. doi: 10.3934/jimo.2020087

[6]

A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35

[7]

Joelma Azevedo, Juan Carlos Pozo, Arlúcio Viana. Global solutions to the non-local Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021146

[8]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[9]

Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768

[10]

Anouar Bahrouni. Trudinger-Moser type inequality and existence of solution for perturbed non-local elliptic operators with exponential nonlinearity. Communications on Pure & Applied Analysis, 2017, 16 (1) : 243-252. doi: 10.3934/cpaa.2017011

[11]

Yang Yang, Kaiyong Wang, Jiajun Liu, Zhimin Zhang. Asymptotics for a bidimensional risk model with two geometric Lévy price processes. Journal of Industrial & Management Optimization, 2019, 15 (2) : 481-505. doi: 10.3934/jimo.2018053

[12]

Xingchun Wang, Yongjin Wang. Hedging strategies for discretely monitored Asian options under Lévy processes. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1209-1224. doi: 10.3934/jimo.2014.10.1209

[13]

Mingshang Hu, Shige Peng. G-Lévy processes under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 1-22. doi: 10.3934/puqr.2021001

[14]

Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255

[15]

Qiyu Jin, Ion Grama, Quansheng Liu. Convergence theorems for the Non-Local Means filter. Inverse Problems & Imaging, 2018, 12 (4) : 853-881. doi: 10.3934/ipi.2018036

[16]

Gabriel Peyré, Sébastien Bougleux, Laurent Cohen. Non-local regularization of inverse problems. Inverse Problems & Imaging, 2011, 5 (2) : 511-530. doi: 10.3934/ipi.2011.5.511

[17]

Olivier Bonnefon, Jérôme Coville, Guillaume Legendre. Concentration phenomenon in some non-local equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 763-781. doi: 10.3934/dcdsb.2017037

[18]

Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741

[19]

Raffaella Servadei, Enrico Valdinoci. A Brezis-Nirenberg result for non-local critical equations in low dimension. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2445-2464. doi: 10.3934/cpaa.2013.12.2445

[20]

Imran H. Biswas, Indranil Chowdhury. On the differentiability of the solutions of non-local Isaacs equations involving $\frac{1}{2}$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 907-927. doi: 10.3934/cpaa.2016.15.907

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (131)
  • HTML views (0)
  • Cited by (32)

Other articles
by authors

[Back to Top]