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January  2022, 15(1): 95-116. doi: 10.3934/dcdss.2021027

Fractional Laplacians : A short survey

1. 

Department of Mathematics and Computer Science, Faculty of sciences Aïn Chock, B.P. 5366 Maarif, Casablanca, Morocco

2. 

Institut Elie Cartan de Lorraine, Université de Lorraine, B.P. 239, Vandoeuvre-lès-Nancy, France

* Corresponding author : El Haj Laamri

Received  August 2020 Revised  December 2020 Published  January 2022 Early access  March 2021

This paper describes the state of the art and gives a survey of the wide literature published in the last years on the fractional Laplacian. We will first recall some definitions of this operator in $ \mathbb{R}^N $ and its main properties. Then, we will introduce the four main operators often used in the case of a bounded domain; and we will give several simple and significant examples to highlight the difference between these four operators. Also we give a rather long list of references : it is certainly not exhaustive but hopefully rich enough to track most connected results. We hope that this short survey will be useful for young researchers of all ages who wish to have a quick idea of the fractional Laplacian(s).

Citation: Maha Daoud, El Haj Laamri. Fractional Laplacians : A short survey. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 95-116. doi: 10.3934/dcdss.2021027
References:
[1]

N. Abatangelo, Large solutions for fractional Laplacian Operators, Ph.D thesis, 2015.

[2]

N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2), (2017), 439–467. doi: 10.1016/j.anihpc.2016.02.001.

[3]

B. Abdellaoui, K. Biroud and E.-H. Laamri, Existence et nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary, To appear in Journal of Evolution Equations.

[4]

G. AcostaJ. P. BorthagarayO. Bruno and M. Maas, Regularity theory and high order numerical methods for the (1d)-fractional Laplacian, Mathematics of Computation, 87 (2018), 1821-1857.  doi: 10.1090/mcom/3276.

[5]

G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM Journal on Numerical Analysis, 55 (2017), 472-495.  doi: 10.1137/15M1033952.

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R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

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B. Alali and N. Albin, Fourier multipliers for nonlocal Laplace operators, Applicable Analysis (2019). doi: 10.1080/00036811.2019.1692134.

[8]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[9]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437.  doi: 10.2140/pjm.1960.10.419.

[10]

U. BiccariM. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.

[11]

U. Biccari, M. Warma and E. Zuazua, Local Regularity for Fractional Heat Equations, Recent Advances in PDEs: Analysis, Numerics and Control, SEMA SIMAI Springer Ser., 17, Springer, Cham, 2018,233–249.

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K. BogdanK. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Rel., 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1.

[15]

K. BogdanT. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, The Annals of Probability, 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.

[16]

M. BonforteA. Figalli and J. L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains, Anal. PDE, 11 (2018), 945-982.  doi: 10.2140/apde.2018.11.945.

[17]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behavior for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

[18]

M. Bonforte and J. L. Vázquez, A Priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Archive for Rational Mechanics and Analysis, 218 (2015), 317-362.  doi: 10.1007/s00205-015-0861-2.

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C. Br$\ddot{a}$ndleE. ColoradoA. De Pablo and U. Sánchez, A concave convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

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B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, Journal of Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

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J. P. Borthagaray and P. Ciarlet, On the convergence in $H^1$-norm for the fractional Laplacian, hal-01912092 (2018). Submitted. doi: 10.1137/18M1221436.

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C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 2016. doi: 10.1007/978-3-319-28739-3.

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X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Physics, 320 (2013), 679-722.  doi: 10.1007/s00220-013-1682-5.

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X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

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L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

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L. Caffarelli; P. R. Stinga :, Fractional elliptic equations, Caccioppoli estimates and regularity., Ann. Inst. H. Poincaré Anal. Non Linéaire 33(3) (2014), 767-807. doi: 10.1016/j.anihpc.2015.01.004.

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X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Problems & Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.

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A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Commun. Part. Differ. Equ., 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954.

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W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Am., 115 (2004), 1424-1430. 

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Z.-Q. ChenP. Kim and R. Song, Two-sided heat kernel estimates for censored stable-like processes, Probab. Theory Rel., 146 (2010), 361-399.  doi: 10.1007/s00440-008-0193-3.

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Z.-Q. Chen and R. Song, Two-sided eigenvalue estimates for subordinate processes in domains, Journal of Functional Analysis, 226 (2005), 90-113.  doi: 10.1016/j.jfa.2005.05.004.

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show all references

References:
[1]

N. Abatangelo, Large solutions for fractional Laplacian Operators, Ph.D thesis, 2015.

[2]

N. Abatangelo and L. Dupaigne, Nonhomogeneous boundary conditions for the spectral fractional Laplacian, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2), (2017), 439–467. doi: 10.1016/j.anihpc.2016.02.001.

[3]

B. Abdellaoui, K. Biroud and E.-H. Laamri, Existence et nonexistence of positive solutions to a fractional parabolic problem with singular weight at the boundary, To appear in Journal of Evolution Equations.

[4]

G. AcostaJ. P. BorthagarayO. Bruno and M. Maas, Regularity theory and high order numerical methods for the (1d)-fractional Laplacian, Mathematics of Computation, 87 (2018), 1821-1857.  doi: 10.1090/mcom/3276.

[5]

G. Acosta and J. P. Borthagaray, A fractional Laplace equation: Regularity of solutions and finite element approximations, SIAM Journal on Numerical Analysis, 55 (2017), 472-495.  doi: 10.1137/15M1033952.

[6]

R. A. Adams and J. F. Fournier, Sobolev Spaces, 2nd edition, Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.

[7]

B. Alali and N. Albin, Fourier multipliers for nonlocal Laplace operators, Applicable Analysis (2019). doi: 10.1080/00036811.2019.1692134.

[8]

D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.

[9]

A. V. Balakrishnan, Fractional powers of closed operators and the semigroups generated by them, Pacific J. Math., 10 (1960), 419-437.  doi: 10.2140/pjm.1960.10.419.

[10]

U. BiccariM. Warma and E. Zuazua, Local elliptic regularity for the Dirichlet fractional Laplacian, Adv. Nonlinear Stud., 17 (2017), 387-409.  doi: 10.1515/ans-2017-0014.

[11]

U. Biccari, M. Warma and E. Zuazua, Local Regularity for Fractional Heat Equations, Recent Advances in PDEs: Analysis, Numerics and Control, SEMA SIMAI Springer Ser., 17, Springer, Cham, 2018,233–249.

[12] G. M. BisciV. D. Radulescu and R. Servadei, Variational methods for nonlocal fractional problems, Cambridge University Press, 2016.  doi: 10.1017/CBO9781316282397.
[13]

S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U. S. A., 35 (1949), 368-370.  doi: 10.1073/pnas.35.7.368.

[14]

K. BogdanK. Burdzy and Z.-Q. Chen, Censored stable processes, Probab. Theory Rel., 127 (2003), 89-152.  doi: 10.1007/s00440-003-0275-1.

[15]

K. BogdanT. Grzywny and M. Ryznar, Heat kernel estimates for the fractional Laplacian with Dirichlet conditions, The Annals of Probability, 38 (2010), 1901-1923.  doi: 10.1214/10-AOP532.

[16]

M. BonforteA. Figalli and J. L. Vázquez, Sharp global estimates for local and nonlocal porous medium-type equations in bounded domains, Anal. PDE, 11 (2018), 945-982.  doi: 10.2140/apde.2018.11.945.

[17]

M. BonforteY. Sire and J. L. Vázquez, Existence, uniqueness and asymptotic behavior for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.  doi: 10.3934/dcds.2015.35.5725.

[18]

M. Bonforte and J. L. Vázquez, A Priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Archive for Rational Mechanics and Analysis, 218 (2015), 317-362.  doi: 10.1007/s00205-015-0861-2.

[19]

C. Br$\ddot{a}$ndleE. ColoradoA. De Pablo and U. Sánchez, A concave convex elliptic problem involving the fractional Laplacian, Proc. R. Soc. Edinb., 143 (2013), 39-71.  doi: 10.1017/S0308210511000175.

[20]

B. BarriosE. ColoradoA. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, Journal of Differential Equations, 252 (2012), 6133-6162.  doi: 10.1016/j.jde.2012.02.023.

[21]

J. P. Borthagaray and P. Ciarlet, On the convergence in $H^1$-norm for the fractional Laplacian, hal-01912092 (2018). Submitted. doi: 10.1137/18M1221436.

[22]

C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lecture Notes of the Unione Matematica Italiana, 2016. doi: 10.1007/978-3-319-28739-3.

[23]

X. Cabré and J.-M. Roquejoffre, The influence of fractional diffusion in Fisher-KPP equations, Comm. Math. Physics, 320 (2013), 679-722.  doi: 10.1007/s00220-013-1682-5.

[24]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53.  doi: 10.1016/j.anihpc.2013.02.001.

[25]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.  doi: 10.1080/03605300600987306.

[26]

L. Caffarelli; P. R. Stinga :, Fractional elliptic equations, Caccioppoli estimates and regularity., Ann. Inst. H. Poincaré Anal. Non Linéaire 33(3) (2014), 767-807. doi: 10.1016/j.anihpc.2015.01.004.

[27]

X. CaoY.-H. Lin and H. Liu, Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators, Inverse Problems & Imaging, 13 (2019), 197-210.  doi: 10.3934/ipi.2019011.

[28]

A. CapellaJ. DávilaL. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local semilinear equations, Commun. Part. Differ. Equ., 36 (2011), 1353-1384.  doi: 10.1080/03605302.2011.562954.

[29]

W. Chen and S. Holm, Fractional Laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency, J. Acoust. Soc. Am., 115 (2004), 1424-1430. 

[30]

Z.-Q. ChenP. Kim and R. Song, Two-sided heat kernel estimates for censored stable-like processes, Probab. Theory Rel., 146 (2010), 361-399.  doi: 10.1007/s00440-008-0193-3.

[31]

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Figure 1.  Comparison of the functions $ (-\Delta)^su $ where $ u $ is defined in (4.11) and $ (-\Delta)^s $ represents $ (-\Delta)_{Spec}^{s} $ (——), $ (-\Delta)_{Reg}^{s} $ (- - - -), $ (-\Delta)_{Rest}^{s} $ (- $ \cdot $ - $ \cdot $) or $ (-\Delta)_{Pery}^{s} $ ($ \cdot $$ \cdot $$ \cdot $$ \cdot $$ \cdot $$ \cdot $) with $ \delta = 4 $. The result for $ -\Delta $ ($ \ast $ $ \ast $ $ \ast $ $ \ast $) is included in the plot of $ s = 0.975 $
Figure 2.  Comparison of the functions $ (-\Delta)^su $ where $ u $ is defined in (4.12) and $ (-\Delta)^s $ represents $ (-\Delta)_{Spec}^{s} $ (——), $ (-\Delta)_{Reg}^{s} $ (- - - -), $ (-\Delta)_{Rest}^{s} $ (- $ \cdot $ - $ \cdot $) or $ (-\Delta)_{Pery}^{s} $ ($ \cdot $$ \cdot $$ \cdot $$ \cdot $$ \cdot $$ \cdot $) with $ \delta = 4 $. The result for $ -\Delta $ ($ \ast $ $ \ast $ $ \ast $ $ \ast $) is included in the plot of $ s = 0.975 $
Figure 3.  Comparison of the solution to (4.13) with $ (-\Delta)_{Spec}^{s} $ (——), $ (-\Delta)_{Reg}^{s} $ (- - - -), $ (-\Delta)_{Rest}^{s} $ (- $ \cdot $ - $ \cdot $) or $ (-\Delta)_{Pery}^{s} $ ($ \cdot $$ \cdot $$ \cdot $$ \cdot $$ \cdot $$ \cdot $) with $ \delta = 4 $. The result for $ -\Delta $ ($ \ast $ $ \ast $ $ \ast $ $ \ast $) is included in the plot of $ s = 0.975 $
Table 1.  The regional fractional Laplacian of some functions that vanish in $ \mathbb{R}\setminus (-1,1) $
$ u(x) $ $ (-\Delta)_{Reg}^su(x) $
$ (1-x^2)_+^{s-1} $ 0
$ (1-x^2)_+^s $ $ \Gamma(2s+1) $
$ (1-x^2)_+^{s+1} $ $ (s+1)\Gamma(2s+1)(1-(2s+1)x^2) $
$ (1-x^2)_+^{s+2} $ $ \frac{(s+1)(s+2)}{2}\Gamma(2s+1)(1-(4s+2)x^2+(\frac{2s}{3}+1)(2s+1)x^4) $
$ x(1-x^2)_+^{s-1} $ 0
$ x(1-x^2)_+^s $ $ \Gamma(2s+2)x $
$ x(1-x^2)_+^{s+1} $ $ \frac{\Gamma(2s+3)}{6}(3-(2s+3)x^2)x $
$ x(1-x^2)_+^{s+2} $ $ \frac{s+2}{60}\Gamma(2s+3)(15-(20s+30)x^2+(2s+3)(2s+5)x^4)x $
$ u(x) $ $ (-\Delta)_{Reg}^su(x) $
$ (1-x^2)_+^{s-1} $ 0
$ (1-x^2)_+^s $ $ \Gamma(2s+1) $
$ (1-x^2)_+^{s+1} $ $ (s+1)\Gamma(2s+1)(1-(2s+1)x^2) $
$ (1-x^2)_+^{s+2} $ $ \frac{(s+1)(s+2)}{2}\Gamma(2s+1)(1-(4s+2)x^2+(\frac{2s}{3}+1)(2s+1)x^4) $
$ x(1-x^2)_+^{s-1} $ 0
$ x(1-x^2)_+^s $ $ \Gamma(2s+2)x $
$ x(1-x^2)_+^{s+1} $ $ \frac{\Gamma(2s+3)}{6}(3-(2s+3)x^2)x $
$ x(1-x^2)_+^{s+2} $ $ \frac{s+2}{60}\Gamma(2s+3)(15-(20s+30)x^2+(2s+3)(2s+5)x^4)x $
Table 2.  The regional fractional Laplacian of some functions that vanish in $ \mathbb{R}^N\setminus B(0,1) $
$ u(\mathbf{x}) $ $ (-\Delta)_{Reg}^su(\mathbf{x}) $
$ (1-\|\mathbf{x}\|^2)_+^s $ $ 4^s\Gamma(s+1)\Gamma(\frac{2s+N}{2})\Gamma(\frac{N}{2} )^{-1} $
$ (1-\|\mathbf{x}\|^2)_+^{s+1} $ $ 4^s\Gamma(s+2)\Gamma(\frac{2s+N}{2}) \Gamma(\frac{N}{2} )^{-1}(1-(1+\frac{2s}{N})\|\mathbf{x}\|^2) $
$ (1-\|\mathbf{x}\|^2)_+^{s}x_N $ $ 4^s\Gamma(s+1)\Gamma(\frac{2s+N}{2}+1)\Gamma(\frac{N}{2}+1 )^{-1}x_N $
$ (1-\|\mathbf{x}\|^2)_+^{s+1}x_N $ $ 4^s\Gamma(s+2)\Gamma(\frac{2s+N}{2}+1)\Gamma(\frac{N}{2}+1 )^{-1}(1-(1+\frac{2s}{N+2})\|\mathbf{x}\|^2) $
$ u(\mathbf{x}) $ $ (-\Delta)_{Reg}^su(\mathbf{x}) $
$ (1-\|\mathbf{x}\|^2)_+^s $ $ 4^s\Gamma(s+1)\Gamma(\frac{2s+N}{2})\Gamma(\frac{N}{2} )^{-1} $
$ (1-\|\mathbf{x}\|^2)_+^{s+1} $ $ 4^s\Gamma(s+2)\Gamma(\frac{2s+N}{2}) \Gamma(\frac{N}{2} )^{-1}(1-(1+\frac{2s}{N})\|\mathbf{x}\|^2) $
$ (1-\|\mathbf{x}\|^2)_+^{s}x_N $ $ 4^s\Gamma(s+1)\Gamma(\frac{2s+N}{2}+1)\Gamma(\frac{N}{2}+1 )^{-1}x_N $
$ (1-\|\mathbf{x}\|^2)_+^{s+1}x_N $ $ 4^s\Gamma(s+2)\Gamma(\frac{2s+N}{2}+1)\Gamma(\frac{N}{2}+1 )^{-1}(1-(1+\frac{2s}{N+2})\|\mathbf{x}\|^2) $
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