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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  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 & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021027
References:
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show all references

References:
[1]

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

[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.  Google Scholar

[3]

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[4]

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[5]

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

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[10]

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[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.  Google Scholar

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[13]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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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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