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.

[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.

[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.

[4]

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

[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.

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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.

[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.

[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.

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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.

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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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[39]

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

[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.

[41]

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

[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.

[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.

[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.

[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.

[46]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[54]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

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

[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.

[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.

[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.

[18]

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

[19]

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

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[39]

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

[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.

[41]

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

[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.

[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.

[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.

[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.

[46]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[54]

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

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