March  2019, 39(3): 1405-1456. doi: 10.3934/dcds.2019061

Regularity estimates for nonlocal Schrödinger equations

African Institute for Mathematical Sciences (AIMS), KM 2, Route de Joal, B.P. 14 18. Mbour, Sénégal

Received  May 2018 Revised  October 2018 Published  December 2018

Fund Project: The author's work is supported by the Alexander von Humboldt foundation. Part of this work was done while he was visiting the Goethe University in Frankfurt am Main during AugustSeptember 2017 and he thanks the Mathematics department for the kind hospitality. The author is grateful to Xavier Ros-Oton, Tobias Weth and Enrico Valdinoci for their availability and for the many useful discussions during the preparation of this work

We are concerned with Hölder regularity estimates for weak solutions $u$ to nonlocal Schrödinger equations subject to exterior Dirichlet conditions in an open set $\Omega\subset \mathbb{R}^N$. The class of nonlocal operators considered here are defined, via Dirichlet forms, by symmetric kernels $K(x, y)$ bounded from above and below by $|x-y|^{-N-2s}$, with $s\in (0, 1)$. The entries in the equations are in some Morrey spaces and the domain $\Omega$ satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When $K$ defines a nonlocal operator with sufficiently regular coefficients, we obtain Hölder estimates, up to the boundary of $ \Omega$, for $u$ and the ratio $u/d^s$, with $d(x) = \text{dist}(x, \mathbb{R}^N\setminus\Omega)$. If the kernel $K$ defines a nonlocal operator with Hölder continuous coefficients and the entries are Hölder continuous, we obtain interior $C^{2s+\beta}$ regularity estimates of the weak solutions $u$. Our argument is based on blow-up analysis and compact Sobolev embedding.

Citation: Mouhamed Moustapha Fall. Regularity estimates for nonlocal Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1405-1456. doi: 10.3934/dcds.2019061
References:
[1]

G. BarlesE. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order elliptic integro-differential equations, J. Eur. Math. Soc., 13 (2011), 1-26. doi: 10.4171/JEMS/242. Google Scholar

[2]

G. BarlesE. Chasseigne and C. Imbert, The Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246. doi: 10.1512/iumj.2008.57.3315. Google Scholar

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B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609–639. Google Scholar

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

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K. BogdanT. Kumagai and M. Kwasnicki, Boundary Harnack inequality for Markov processes with jumps, Trans. Amer. Math. Soc., 367 (2015), 477-517. doi: 10.1090/S0002-9947-2014-06127-8. Google Scholar

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K. Bogdan and T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Math., 133 (1999), 53-92. doi: 10.4064/sm-133-1-53-92. Google Scholar

[7]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80. doi: 10.4064/sm-123-1-43-80. Google Scholar

[8]

L. CaffarelliX. Ros-Oton and J. Serra, Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries, Invent. Math., 208 (2017), 1155-1211. doi: 10.1007/s00222-016-0703-3. Google Scholar

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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

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L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Rat. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4. Google Scholar

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L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math., 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9. Google Scholar

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L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213. doi: 10.2307/1971480. Google Scholar

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Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232. Google Scholar

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Y. Chen, Regularity of the solution to the Dirichlet problem in Morrey spaces, J. Partial Differential Equations, 15 (2002), 37-46. Google Scholar

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M. CostabelM. Dauge and R. Duduchava, Asymptotics without logarithmic terms for crack problems, Comm. Partial Differential Equations, 28 (2003), 869-926. doi: 10.1081/PDE-120021180. Google Scholar

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G. Di Fazio, Hölder continuity of solutions for some Schrödinger equations, Rend. Sem. Mat. Univ. di Padova, 79 (1988), 173-183. Google Scholar

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M. M. Fall and X. Ros-Oton, Nonlocal Schrödinger equations with potentials in Kato class, Forthcoming.Google Scholar

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M. M. Fall, Regularity results for nonlocal equations and applications, https://arXiv.org/abs/1806.09139.Google Scholar

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M. M. Fall and T. Weth, Liouville theorems for a general class of nonlocal operators, Potential Anal., 45 (2016), 187-200. doi: 10.1007/s11118-016-9546-1. Google Scholar

[22]

M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3. Google Scholar

[23]

X. Fernández-Real and X. Ros-Oton, Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221. doi: 10.1016/j.jfa.2017.02.015. Google Scholar

[24]

X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. Acad. Cienc. Ser. A Math., 110 (2016), 49-64. doi: 10.1007/s13398-015-0218-6. Google Scholar

[25]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. Google Scholar

[26]

L. Geisinger, A short proof of Weyl's law for fractional differential operators, J. Math. Phys., 55 (2014), 011504, 7pp. doi: 10.1063/1.4861935. Google Scholar

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D. Gilbarg and N. Trudinger, l S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. Google Scholar

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W. B. Gordon, On the diffeomorphisms of Euclidean space, Amer. Math. Monthly, 79 (1972), 755-759. doi: 10.1080/00029890.1972.11993118. Google Scholar

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L. Grafakos, Modern Fourier Analysis, Second edition. Graduate Texts in Mathematics, 250. Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2. Google Scholar

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[31]

G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE, 7 (2014), 1649-1682. doi: 10.2140/apde.2014.7.1649. Google Scholar

[32]

G. Grubb, Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Differential Equations, 261 (2016), 1835-1879. doi: 10.1016/j.jde.2016.04.017. Google Scholar

[33]

W. Hoh and N. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Funct. Anal., 137 (1996), 19-48. doi: 10.1006/jfan.1996.0039. Google Scholar

[34]

T. Jin and J. Xiong, Schauder estimates for nonlocal fully nonlinear equations, Ann. Inst. H. Poincaré Abal. Non Linnéaire, 33 (2016), 1375-1407. doi: 10.1016/j.anihpc.2015.05.004. Google Scholar

[35]

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

[36]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators Ⅱ, preprint, https://arXiv.org/abs/1412.7566.Google Scholar

[37]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators, J. Eur. Math. Soc. (JEMS), 19 (2017), 983-1011. doi: 10.4171/JEMS/686. Google Scholar

[38]

M. KassmannM. Rang and R. W. Schwab, Integro-differential equations with nonlinear directional dependence, Indiana Univ. Math. J., 63 (2014), 1467-1498. doi: 10.1512/iumj.2014.63.5394. Google Scholar

[39]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212. Google Scholar

[40]

D. Kriventsov, C1, α interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations, 38 (2013), 2081-2106. doi: 10.1080/03605302.2013.831990. Google Scholar

[41]

M. Kassmann, Analysis of symmetric Markov processes, A localization technique for non-local operators. Universität Bonn, Habilitation Thesis, 2007.Google Scholar

[42]

T. KuusiG. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2. Google Scholar

[43]

R. Monneau, Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients, Journal of Fourier Analysis and Applications, 15 (2009), 279-335. doi: 10.1007/s00041-009-9066-0. Google Scholar

[44]

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166. doi: 10.1090/S0002-9947-1938-1501936-8. Google Scholar

[45]

X. Ros-Oton and J. Serra, Boundary regularity estimates for nonlocal elliptic equations in C1 and C1, α domains, Ann. Mat. Pura Appl.(4), 196 (2017), 1637-1668. doi: 10.1007/s10231-016-0632-1. Google Scholar

[46]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. doi: 10.5565/PUBLMAT_60116_01. Google Scholar

[47]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Differential Equations, 260 (2016), 8675-8715. doi: 10.1016/j.jde.2016.02.033. Google Scholar

[48]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J., 165 (2016), 2079-2154. doi: 10.1215/00127094-3476700. Google Scholar

[49]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl.(9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar

[50]

R. W. Schwab and L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, 9 (2016), 727-772. doi: 10.2140/apde.2016.9.727. Google Scholar

[51]

J. Serra, Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations, 54 (2015), 3571-3601. doi: 10.1007/s00526-015-0914-2. Google Scholar

[52]

J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations, 54 (2015), 615-629. doi: 10.1007/s00526-014-0798-6. Google Scholar

[53]

R. Song and J.-M. Wu, Boundary Harnack Principle for Symmetric Stable Processes, J. Func. Anal., 168 (1999), 403-427. doi: 10.1006/jfan.1999.3470. Google Scholar

[54]

E. Stein, Singular integrals and differentiability properties of functions, Princeton, University Press, 1970. Google Scholar

show all references

References:
[1]

G. BarlesE. Chasseigne and C. Imbert, Hölder continuity of solutions of second-order elliptic integro-differential equations, J. Eur. Math. Soc., 13 (2011), 1-26. doi: 10.4171/JEMS/242. Google Scholar

[2]

G. BarlesE. Chasseigne and C. Imbert, The Dirichlet problem for second-order elliptic integro-differential equations, Indiana Univ. Math. J., 57 (2008), 213-246. doi: 10.1512/iumj.2008.57.3315. Google Scholar

[3]

B. Barrios, A. Figalli and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 609–639. Google Scholar

[4]

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

[5]

K. BogdanT. Kumagai and M. Kwasnicki, Boundary Harnack inequality for Markov processes with jumps, Trans. Amer. Math. Soc., 367 (2015), 477-517. doi: 10.1090/S0002-9947-2014-06127-8. Google Scholar

[6]

K. Bogdan and T. Byczkowski, Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains, Studia Math., 133 (1999), 53-92. doi: 10.4064/sm-133-1-53-92. Google Scholar

[7]

K. Bogdan, The boundary Harnack principle for the fractional Laplacian, Studia Math., 123 (1997), 43-80. doi: 10.4064/sm-123-1-43-80. Google Scholar

[8]

L. CaffarelliX. Ros-Oton and J. Serra, Obstacle problems for integro-differential operators: Regularity of solutions and free boundaries, Invent. Math., 208 (2017), 1155-1211. doi: 10.1007/s00222-016-0703-3. Google Scholar

[9]

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

[10]

L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Rat. Mech. Anal., 200 (2011), 59-88. doi: 10.1007/s00205-010-0336-4. Google Scholar

[11]

L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for nonlocal fully nonlinear equations, Ann. of Math., 174 (2011), 1163-1187. doi: 10.4007/annals.2011.174.2.9. Google Scholar

[12]

L. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math.(2), 130 (1989), 189-213. doi: 10.2307/1971480. Google Scholar

[13]

Z.-Q. Chen and R. Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann., 312 (1998), 465-501. doi: 10.1007/s002080050232. Google Scholar

[14]

Y. Chen, Regularity of the solution to the Dirichlet problem in Morrey spaces, J. Partial Differential Equations, 15 (2002), 37-46. Google Scholar

[15]

M. CostabelM. Dauge and R. Duduchava, Asymptotics without logarithmic terms for crack problems, Comm. Partial Differential Equations, 28 (2003), 869-926. doi: 10.1081/PDE-120021180. Google Scholar

[16]

M. Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes, J. Funct. Anal., 272 (2017), 4762-4837. doi: 10.1016/j.jfa.2017.02.016. Google Scholar

[17]

A. Di CastroT. Kuusi and G. Palatucci, Local behavior of fractional p-minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire. V., 33 (2016), 1279-1299. doi: 10.1016/j.anihpc.2015.04.003. Google Scholar

[18]

G. Di Fazio, Hölder continuity of solutions for some Schrödinger equations, Rend. Sem. Mat. Univ. di Padova, 79 (1988), 173-183. Google Scholar

[19]

M. M. Fall and X. Ros-Oton, Nonlocal Schrödinger equations with potentials in Kato class, Forthcoming.Google Scholar

[20]

M. M. Fall, Regularity results for nonlocal equations and applications, https://arXiv.org/abs/1806.09139.Google Scholar

[21]

M. M. Fall and T. Weth, Liouville theorems for a general class of nonlocal operators, Potential Anal., 45 (2016), 187-200. doi: 10.1007/s11118-016-9546-1. Google Scholar

[22]

M. FelsingerM. Kassmann and P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3. Google Scholar

[23]

X. Fernández-Real and X. Ros-Oton, Regularity theory for general stable operators: Parabolic equations, J. Funct. Anal., 272 (2017), 4165-4221. doi: 10.1016/j.jfa.2017.02.015. Google Scholar

[24]

X. Fernández-Real and X. Ros-Oton, Boundary regularity for the fractional heat equation, Rev. Acad. Cienc. Ser. A Math., 110 (2016), 49-64. doi: 10.1007/s13398-015-0218-6. Google Scholar

[25]

G. Franzina and G. Palatucci, Fractional p-eigenvalues, Riv. Mat. Univ. Parma, 5 (2014), 373-386. Google Scholar

[26]

L. Geisinger, A short proof of Weyl's law for fractional differential operators, J. Math. Phys., 55 (2014), 011504, 7pp. doi: 10.1063/1.4861935. Google Scholar

[27]

D. Gilbarg and N. Trudinger, l S. Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer-Verlag, Berlin, 2001. xiv+517 pp. Google Scholar

[28]

W. B. Gordon, On the diffeomorphisms of Euclidean space, Amer. Math. Monthly, 79 (1972), 755-759. doi: 10.1080/00029890.1972.11993118. Google Scholar

[29]

L. Grafakos, Modern Fourier Analysis, Second edition. Graduate Texts in Mathematics, 250. Springer, New York, 2009. doi: 10.1007/978-0-387-09434-2. Google Scholar

[30]

G. Grubb, Fractional Laplacians on domains, a development of Hormander's theory of μ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. Google Scholar

[31]

G. Grubb, Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators, Anal. PDE, 7 (2014), 1649-1682. doi: 10.2140/apde.2014.7.1649. Google Scholar

[32]

G. Grubb, Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Differential Equations, 261 (2016), 1835-1879. doi: 10.1016/j.jde.2016.04.017. Google Scholar

[33]

W. Hoh and N. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Funct. Anal., 137 (1996), 19-48. doi: 10.1006/jfan.1996.0039. Google Scholar

[34]

T. Jin and J. Xiong, Schauder estimates for nonlocal fully nonlinear equations, Ann. Inst. H. Poincaré Abal. Non Linnéaire, 33 (2016), 1375-1407. doi: 10.1016/j.anihpc.2015.05.004. Google Scholar

[35]

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

[36]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators Ⅱ, preprint, https://arXiv.org/abs/1412.7566.Google Scholar

[37]

M. Kassmann and A. Mimica, Intrinsic scaling properties for nonlocal operators, J. Eur. Math. Soc. (JEMS), 19 (2017), 983-1011. doi: 10.4171/JEMS/686. Google Scholar

[38]

M. KassmannM. Rang and R. W. Schwab, Integro-differential equations with nonlinear directional dependence, Indiana Univ. Math. J., 63 (2014), 1467-1498. doi: 10.1512/iumj.2014.63.5394. Google Scholar

[39]

M. Kassmann and R. W. Schwab, Regularity results for nonlocal parabolic equations, Riv. Math. Univ. Parma (N.S.), 5 (2014), 183-212. Google Scholar

[40]

D. Kriventsov, C1, α interior regularity for nonlinear nonlocal elliptic equations with rough kernels, Comm. Partial Differential Equations, 38 (2013), 2081-2106. doi: 10.1080/03605302.2013.831990. Google Scholar

[41]

M. Kassmann, Analysis of symmetric Markov processes, A localization technique for non-local operators. Universität Bonn, Habilitation Thesis, 2007.Google Scholar

[42]

T. KuusiG. Mingione and Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317-1368. doi: 10.1007/s00220-015-2356-2. Google Scholar

[43]

R. Monneau, Pointwise Estimates for Laplace Equation. Applications to the Free Boundary of the Obstacle Problem with Dini Coefficients, Journal of Fourier Analysis and Applications, 15 (2009), 279-335. doi: 10.1007/s00041-009-9066-0. Google Scholar

[44]

C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166. doi: 10.1090/S0002-9947-1938-1501936-8. Google Scholar

[45]

X. Ros-Oton and J. Serra, Boundary regularity estimates for nonlocal elliptic equations in C1 and C1, α domains, Ann. Mat. Pura Appl.(4), 196 (2017), 1637-1668. doi: 10.1007/s10231-016-0632-1. Google Scholar

[46]

X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. doi: 10.5565/PUBLMAT_60116_01. Google Scholar

[47]

X. Ros-Oton and J. Serra, Regularity theory for general stable operators, J. Differential Equations, 260 (2016), 8675-8715. doi: 10.1016/j.jde.2016.02.033. Google Scholar

[48]

X. Ros-Oton and J. Serra, Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J., 165 (2016), 2079-2154. doi: 10.1215/00127094-3476700. Google Scholar

[49]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl.(9), 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar

[50]

R. W. Schwab and L. Silvestre, Regularity for parabolic integro-differential equations with very irregular kernels, Anal. PDE, 9 (2016), 727-772. doi: 10.2140/apde.2016.9.727. Google Scholar

[51]

J. Serra, Cσ+α regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations, 54 (2015), 3571-3601. doi: 10.1007/s00526-015-0914-2. Google Scholar

[52]

J. Serra, Regularity for fully nonlinear nonlocal parabolic equations with rough kernels, Calc. Var. Partial Differential Equations, 54 (2015), 615-629. doi: 10.1007/s00526-014-0798-6. Google Scholar

[53]

R. Song and J.-M. Wu, Boundary Harnack Principle for Symmetric Stable Processes, J. Func. Anal., 168 (1999), 403-427. doi: 10.1006/jfan.1999.3470. Google Scholar

[54]

E. Stein, Singular integrals and differentiability properties of functions, Princeton, University Press, 1970. Google Scholar

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