Limited regularity of solutions to fractional heat and Schrödinger equations

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June 2019, 39(6): 3609-3634. doi: 10.3934/dcds.2019148

Limited regularity of solutions to fractional heat and Schrödinger equations

Gerd Grubb

Department of Mathematical Sciences, Copenhagen University, Universitetsparken 5, DK-2100 Copenhagen, Denmark

Received October 2018 Revised December 2018 Published February 2019

When $ P$ is the fractional Laplacian $ (-\Delta )^a$, $ 0<a<1$, or a pseudodifferential generalization thereof, the Dirichlet problem for the associated heat equation over a smooth set $ \Omega \subset{\Bbb R}^n$: $ r^+Pu(x, t)+\partial_tu(x, t) = f(x, t)$ on $ \Omega \times \, ]0, T[\, $, $ u(x, t) = 0$ for $ x\notin\Omega$, $ u(x, 0) = 0$, is known to be solvable in relatively low-order Sobolev or Hölder spaces. We now show that in contrast with differential operator cases, the regularity of $ u$ in $ x$ at $ \partial\Omega$ when $ f$ is very smooth cannot in general be improved beyond a certain estimate. An improvement requires the vanishing of a Neumann boundary value. --- There is a similar result for the Schrödinger Dirichlet problem $ r^+Pv(x)+Vv(x) = g(x)$ on $ \Omega$, $ \text{supp } v\subset\overline\Omega$, with $ V(x)\in C^\infty$. The proofs involve a precise description, of interest in itself, of the Dirichlet domains in terms of regular functions and functions with a $ \text{dist}(x, \partial\Omega )^a$ singularity.

Keywords: Fractional Laplacian, stable process, pseudodifferential operator, fractional heat equation, fractional Schrödinger Dirichlet problem, Lp and Hölder estimates, limited spatial regularity.

Mathematics Subject Classification: Primary: 35K05, 35K35; Secondary: 35S11, 47G30, 60G52.

Citation: Gerd Grubb. Limited regularity of solutions to fractional heat and Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3609-3634. doi: 10.3934/dcds.2019148

References:

[1]	N. Abatangelo, Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607. doi: 10.3934/dcds.2015.35.5555. Google Scholar
[2]	N. Abatangelo, S. Dipierro, M. M. Fall, S. Jarohs and A. Saldana, Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions, arXiv: 1806.05128.Google Scholar
[3]	N. Abatangelo, S. Jarohs and A. Saldana, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Contemp. Math., 20 (2018), 1850002, 36 pp, arXiv: 1707.03603. doi: 10.1142/S0219199718500025. Google Scholar
[4]	H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102. Google Scholar
[5]	U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, arXiv: 1704.07562.Google Scholar
[6]	B. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408. Google Scholar
[7]	K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes, Prob. Theory Related Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1. Google Scholar
[8]	M. Bonforte, Y. Sire and J. L. Vazquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767. doi: 10.3934/dcds.2015.35.5725. Google Scholar
[9]	L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11-51. doi: 10.1007/BF02392024. Google Scholar
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar
[11]	H. Chang-Lara and G. Davila, Regularity for solutions of non local parabolic equations, Calc. Var. Part. Diff. Equations, 49 (2014), 139-172. doi: 10.1007/s00526-012-0576-2. Google Scholar
[12]	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
[13]	R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. Google Scholar
[14]	R. Courant and D. Hilbert, Methods of Mathematical Physics II, Interscience Publishers, New York, 1962.Google Scholar
[15]	J. I. Diaz, D. Gomez-Castro and J. L. Vazquez, The fractional Schrödinger equation with general nonnegative potentials, The weighted space approach, arXiv: 1804.08398.Google Scholar
[16]	M. M. Fall, Regularity estimates for nonlocal Schrödinger equations, arXiv: 1711.02206.Google Scholar
[17]	M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Part. Diff. Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211. Google Scholar
[18]	M. Felsinger, M. 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
[19]	X. Fernandez-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
[20]	R. Frank and L. Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, J. Reine Angew. Math., 712 (2016), 1-37. doi: 10.1515/crelle-2013-0120. Google Scholar
[21]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv: 1609.09248.Google Scholar
[22]	M. Gonzalez, R. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863. doi: 10.1007/s12220-011-9217-9. Google Scholar
[23]	G. Grubb, Pseudo-differential boundary problems in Lp spaces, Comm. Part. Diff. Eq., 15 (1990), 289-340. doi: 10.1080/03605309908820688. Google Scholar
[24]	G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Progress in Math. vol. 65, Second Edition, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-0769-6. Google Scholar
[25]	G. Grubb, Distributions and Operators. Graduate Texts in Mathematics, 252, Springer, New York, 2009. Google Scholar
[26]	G. Grubb, Local and nonlocal boundary conditions for µ-transmission and fractional elliptic pseudodifferential operators, Analysis and P.D.E., 7 (2014), 1649-1682. doi: 10.2140/apde.2014.7.1649. Google Scholar
[27]	G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. Google Scholar
[28]	G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634. doi: 10.1016/j.jmaa.2014.07.081. Google Scholar
[29]	G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844. doi: 10.1002/mana.201500041. Google Scholar
[30]	G. Grubb, Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Diff. Eq., 261 (2016), 1835-1879. doi: 10.1016/j.jde.2016.04.017. Google Scholar
[31]	G. Grubb, Regularity in L_p Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), 2634-2660. doi: 10.1016/j.jfa.2017.12.011. Google Scholar
[32]	G. Grubb, Green's formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators, Comm. Part. Diff. Equ., 43 (2018), 750–789, arXiv: 1611.03024, https://www.tandfonline.com/doi/full/10.1080/03605302.2018.1475487. doi: 10.1080/03605302.2018.1475487. Google Scholar
[33]	G. Grubb, Fractional-order operators: Boundary problems, heat equations, in Mathematical Analysis and Applications –- Plenary Lectures, ISAAC 2017, Vaxjo Sweden, Springer Proceedings in Mathematics and Statistics, (L. G. Rodino and J. Toft, eds.), Springer, Switzerland, 2018, pp. 51–81.Google Scholar
[34]	W. Hoh and N. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Functional Anal., 137 (1996), 19-48. doi: 10.1006/jfan.1996.0039. Google Scholar
[35]	L. Hörmander, Seminar notes on pseudo-differential operators and boundary problems, Lectures at IAS Princeton 1965-66, available from Lund University, https://lup.lub.lu.se/search/.Google Scholar
[36]	L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer Verlag, Berlin, 1985. Google Scholar
[37]	N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. I–3, Imperial College Press, London, 2001. doi: 10.1142/9781860949746. Google Scholar
[38]	T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22 (2002), 419-441. Google Scholar
[39]	T. Jin and J. Xiong, Schauder estimates for solutions of linear parabolic integro-differential equations, Discrete Contin. Dyn. Syst., 35 (2015), 5977-5998. doi: 10.3934/dcds.2015.35.5977. Google Scholar
[40]	J. Johnsen, Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces, Math. Scand., 79 (1996), 25-85. doi: 10.7146/math.scand.a-12593. Google Scholar
[41]	T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. Google Scholar
[42]	N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, (Translated from the Russian by A. P. Doohovskoy.), Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[43]	T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068. doi: 10.3934/dcds.2015.35.6031. Google Scholar
[44]	R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790. doi: 10.1080/03605302.2013.864304. Google Scholar
[45]	X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. Google Scholar
[46]	X. Ros-Oton, Boundary regularity, Pohozaev identities and nonexistence results, Recent Developments in Nonlocal Theory, De Gruyter, Berlin, 2018, pp. 335–358. Google Scholar
[47]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar
[48]	X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628. doi: 10.1007/s00205-014-0740-2. Google Scholar
[49]	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
[50]	X. Ros-Oton and H. Vivas, Higher-order boundary regularity estimates for nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 111, 20 pp. doi: 10.1007/s00526-018-1399-6. Google Scholar
[51]	R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. Google Scholar
[52]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar
[53]	M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981. Google Scholar

show all references

References:

[1]	N. Abatangelo, Large s-harmonic functions and boundary blow-up solutions for the fractional Laplacian, Discrete Contin. Dyn. Syst., 35 (2015), 5555-5607. doi: 10.3934/dcds.2015.35.5555. Google Scholar
[2]	N. Abatangelo, S. Dipierro, M. M. Fall, S. Jarohs and A. Saldana, Positive powers of the Laplacian in the half-space under Dirichlet boundary conditions, arXiv: 1806.05128.Google Scholar
[3]	N. Abatangelo, S. Jarohs and A. Saldana, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Contemp. Math., 20 (2018), 1850002, 36 pp, arXiv: 1707.03603. doi: 10.1142/S0219199718500025. Google Scholar
[4]	H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr., 186 (1997), 5-56. doi: 10.1002/mana.3211860102. Google Scholar
[5]	U. Biccari, M. Warma and E. Zuazua, Local regularity for fractional heat equations, arXiv: 1704.07562.Google Scholar
[6]	B. M. Blumenthal and R. K. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math., 9 (1959), 399-408. Google Scholar
[7]	K. Bogdan, K. Burdzy and Z.-Q. Chen, Censored stable processes, Prob. Theory Related Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1. Google Scholar
[8]	M. Bonforte, Y. Sire and J. L. Vazquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767. doi: 10.3934/dcds.2015.35.5725. Google Scholar
[9]	L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math., 126 (1971), 11-51. doi: 10.1007/BF02392024. Google Scholar
[10]	L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 (2007), 1245-1260. doi: 10.1080/03605300600987306. Google Scholar
[11]	H. Chang-Lara and G. Davila, Regularity for solutions of non local parabolic equations, Calc. Var. Part. Diff. Equations, 49 (2014), 139-172. doi: 10.1007/s00526-012-0576-2. Google Scholar
[12]	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
[13]	R. Cont and P. Tankov, Financial Modelling with Jump Processes, Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton, FL, 2004. Google Scholar
[14]	R. Courant and D. Hilbert, Methods of Mathematical Physics II, Interscience Publishers, New York, 1962.Google Scholar
[15]	J. I. Diaz, D. Gomez-Castro and J. L. Vazquez, The fractional Schrödinger equation with general nonnegative potentials, The weighted space approach, arXiv: 1804.08398.Google Scholar
[16]	M. M. Fall, Regularity estimates for nonlocal Schrödinger equations, arXiv: 1711.02206.Google Scholar
[17]	M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Comm. Part. Diff. Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211. Google Scholar
[18]	M. Felsinger, M. 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
[19]	X. Fernandez-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
[20]	R. Frank and L. Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, J. Reine Angew. Math., 712 (2016), 1-37. doi: 10.1515/crelle-2013-0120. Google Scholar
[21]	T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv: 1609.09248.Google Scholar
[22]	M. Gonzalez, R. Mazzeo and Y. Sire, Singular solutions of fractional order conformal Laplacians, J. Geom. Anal., 22 (2012), 845-863. doi: 10.1007/s12220-011-9217-9. Google Scholar
[23]	G. Grubb, Pseudo-differential boundary problems in Lp spaces, Comm. Part. Diff. Eq., 15 (1990), 289-340. doi: 10.1080/03605309908820688. Google Scholar
[24]	G. Grubb, Functional Calculus of Pseudodifferential Boundary Problems, Progress in Math. vol. 65, Second Edition, Birkhäuser, Boston, 1996. doi: 10.1007/978-1-4612-0769-6. Google Scholar
[25]	G. Grubb, Distributions and Operators. Graduate Texts in Mathematics, 252, Springer, New York, 2009. Google Scholar
[26]	G. Grubb, Local and nonlocal boundary conditions for µ-transmission and fractional elliptic pseudodifferential operators, Analysis and P.D.E., 7 (2014), 1649-1682. doi: 10.2140/apde.2014.7.1649. Google Scholar
[27]	G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of µ-transmission pseudodifferential operators, Adv. Math., 268 (2015), 478-528. doi: 10.1016/j.aim.2014.09.018. Google Scholar
[28]	G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634. doi: 10.1016/j.jmaa.2014.07.081. Google Scholar
[29]	G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844. doi: 10.1002/mana.201500041. Google Scholar
[30]	G. Grubb, Integration by parts and Pohozaev identities for space-dependent fractional-order operators, J. Diff. Eq., 261 (2016), 1835-1879. doi: 10.1016/j.jde.2016.04.017. Google Scholar
[31]	G. Grubb, Regularity in L_p Sobolev spaces of solutions to fractional heat equations, J. Funct. Anal., 274 (2018), 2634-2660. doi: 10.1016/j.jfa.2017.12.011. Google Scholar
[32]	G. Grubb, Green's formula and a Dirichlet-to-Neumann operator for fractional-order pseudodifferential operators, Comm. Part. Diff. Equ., 43 (2018), 750–789, arXiv: 1611.03024, https://www.tandfonline.com/doi/full/10.1080/03605302.2018.1475487. doi: 10.1080/03605302.2018.1475487. Google Scholar
[33]	G. Grubb, Fractional-order operators: Boundary problems, heat equations, in Mathematical Analysis and Applications –- Plenary Lectures, ISAAC 2017, Vaxjo Sweden, Springer Proceedings in Mathematics and Statistics, (L. G. Rodino and J. Toft, eds.), Springer, Switzerland, 2018, pp. 51–81.Google Scholar
[34]	W. Hoh and N. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Functional Anal., 137 (1996), 19-48. doi: 10.1006/jfan.1996.0039. Google Scholar
[35]	L. Hörmander, Seminar notes on pseudo-differential operators and boundary problems, Lectures at IAS Princeton 1965-66, available from Lund University, https://lup.lub.lu.se/search/.Google Scholar
[36]	L. Hörmander, The Analysis of Linear Partial Differential Operators, III, Springer Verlag, Berlin, 1985. Google Scholar
[37]	N. Jacob, Pseudo Differential Operators and Markov Processes, Vol. I–3, Imperial College Press, London, 2001. doi: 10.1142/9781860949746. Google Scholar
[38]	T. Jakubowski, The estimates for the Green function in Lipschitz domains for the symmetric stable processes, Probab. Math. Statist., 22 (2002), 419-441. Google Scholar
[39]	T. Jin and J. Xiong, Schauder estimates for solutions of linear parabolic integro-differential equations, Discrete Contin. Dyn. Syst., 35 (2015), 5977-5998. doi: 10.3934/dcds.2015.35.5977. Google Scholar
[40]	J. Johnsen, Elliptic boundary problems and the Boutet de Monvel calculus in Besov and Triebel-Lizorkin spaces, Math. Scand., 79 (1996), 25-85. doi: 10.7146/math.scand.a-12593. Google Scholar
[41]	T. Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist., 17 (1997), 339-364. Google Scholar
[42]	N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180, (Translated from the Russian by A. P. Doohovskoy.), Springer-Verlag, New York-Heidelberg, 1972. Google Scholar
[43]	T. Leonori, I. Peral, A. Primo and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst., 35 (2015), 6031-6068. doi: 10.3934/dcds.2015.35.6031. Google Scholar
[44]	R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Part. Diff. Eq., 39 (2014), 1780-1790. doi: 10.1080/03605302.2013.864304. Google Scholar
[45]	X. Ros-Oton, Nonlocal elliptic equations in bounded domains: A survey, Publ. Mat., 60 (2016), 3-26. Google Scholar
[46]	X. Ros-Oton, Boundary regularity, Pohozaev identities and nonexistence results, Recent Developments in Nonlocal Theory, De Gruyter, Berlin, 2018, pp. 335–358. Google Scholar
[47]	X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003. Google Scholar
[48]	X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628. doi: 10.1007/s00205-014-0740-2. Google Scholar
[49]	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
[50]	X. Ros-Oton and H. Vivas, Higher-order boundary regularity estimates for nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 111, 20 pp. doi: 10.1007/s00526-018-1399-6. Google Scholar
[51]	R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh, 144 (2014), 831-855. doi: 10.1017/S0308210512001783. Google Scholar
[52]	L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112. doi: 10.1002/cpa.20153. Google Scholar
[53]	M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981. Google Scholar

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