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doi: 10.3934/dcdsb.2021167
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The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations

1. 

Department of Mathematics, National Taiwan University, Taipei 106

2. 

Faculty of Mathematics and Physics, Kanazawa University, Kakuma, Kanazawa 920-1192

* Corresponding author: Pu-Zhao Kow

Received  January 2021 Revised  May 2021 Early access June 2021

Fund Project: This work was partially supported by JSPS KAKENHI JP20H01812, JP20H00117, JP20KK0058, and MOST 108-2115-M-002-002-MY3

In this paper, we consider a Lewy-Stampacchia-type inequality for the fractional Laplacian on a bounded domain in Euclidean space. Using this inequality, we can show the well-posedness of fractional-type anomalous unidirectional diffusion equations. This study is an extension of the work by Akagi-Kimura (2019) for the standard Laplacian. However, there exist several difficulties due to the nonlocal feature of the fractional Laplacian. We overcome those difficulties employing the Caffarelli-Silvestre extension of the fractional Laplacian.

Citation: Pu-Zhao Kow, Masato Kimura. The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021167
References:
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G. Akagi and M. Kimura, Unidirectional evolution equations of diffusion type, J. Differential Equations, 266 (2019), 1-43. doi: 10.1016/j.jde.2018.05.022.  Google Scholar

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H. Antil and C. Rautenberg, Fractional elliptic quasi-variational inequalities: Theory and numerics, Interfaces Free Bound., 20 (2018), 1-24. doi: 10.4171/IFB/395.  Google Scholar

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

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L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807. doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

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S. DuoH. W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., 355 (2018), 233-252.  doi: 10.1016/j.jcp.2017.11.011.  Google Scholar

[12]

S. DuoH. Wang and Y. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110.  Google Scholar

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L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.  Google Scholar

[14]

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

[15]

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M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

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J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 1, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 2, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 182, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65217-2.  Google Scholar

[21]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 3, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 183, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65393-3.  Google Scholar

[22]

C. W. K. Lo and J. F. Rodrigues, On a class of fractional obstacle type problems related to the distributional Riesz derivative, preprint, arXiv: 2101.06863. Google Scholar

[23]

H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math., 22 (1969), 153-188.  doi: 10.1002/cpa.3160220203.  Google Scholar

[24]

W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press. Cambridge, 2000.  Google Scholar

[25]

S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342.  doi: 10.1016/j.jmaa.2010.12.027.  Google Scholar

[26]

R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations, 39 (2014), 1780-1790.  doi: 10.1080/03605302.2013.864304.  Google Scholar

[27]

R. Musina and A. I. Nazarov, On fractional Laplacians - 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1667-1673.  doi: 10.1016/j.anihpc.2015.08.001.  Google Scholar

[28]

R. Musina and A. I. Nazarov, Variational inequalities for the spectral fractional Laplacian, Comput. Math. Math. Phys., 57 (2017), 373-386. doi: 10.1134/S0965542517030113.  Google Scholar

[29]

R. Musina, A. I. Nazarov and K. Sreenadh, Variational inequalities for the fractional Laplacian, Potential Anal., 46 (2017), 485-498. doi: 10.1007/s11118-016-9591-9.  Google Scholar

[30]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x.  Google Scholar

[31]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.  Google Scholar

[32]

A. Rüland and J.-N. Wang, On the Fractional Landis Conjecture, J. Funct. Anal., 277 (2019), 3236-3270.  doi: 10.1016/j.jfa.2019.05.026.  Google Scholar

[33]

R. L. Schilling, An introduction to Lévy and Feller processes, in From Lévy-Type Processes to Parabolic SPDEs, by D. Khoshnevisan and R. Schilling (eds. F. Utzet and L. Quer-Sardanyons), 1-126, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-34120-0.  Google Scholar

[34]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/049.  Google Scholar

[35]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[36]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.  doi: 10.4171/RMI/750.  Google Scholar

[37]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4.  Google Scholar

[38]

K. Yosida, Functional Analysis, 6$^{th}$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123. Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/978-3-642-61859-8.  Google Scholar

show all references

References:
[1]

M. Ainsworth and C. Glusa, Towards an efficient finite element method for the integral fractional Laplacian on polygonal domains, in Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan, 1, Springer, Cham, 2018, 17-57, arXiv: 1708.01923.  Google Scholar

[2]

G. Akagi and M. Kimura, Unidirectional evolution equations of diffusion type, J. Differential Equations, 266 (2019), 1-43. doi: 10.1016/j.jde.2018.05.022.  Google Scholar

[3]

H. Antil and C. Rautenberg, Fractional elliptic quasi-variational inequalities: Theory and numerics, Interfaces Free Bound., 20 (2018), 1-24. doi: 10.4171/IFB/395.  Google Scholar

[4]

H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. doi: 10.1007/978-0-387-70914-7.  Google Scholar

[5]

H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, (French) [Maximal monotone operators and semi-groups of contractions in Hilbert spaces], North-Holland Mathematics Studies, 5, Notas de Matemática, 50, North-Holland Publishing Co., Amsterdam-London, American Elsevier Publishing Co., Inc., New York, 1973. doi: 10.1016/s0304-0208(08)x7125-7.  Google Scholar

[6]

S. N. Chandler-Wilde, D. P. Hewett and A. Moiola, Sobolev spaces on non-Lipschitz subsets of $\mathbb{R}^{n}$ with application to boundary integral equations on fractal screens, Integral Equations Operator Theory, 87 (2017), 179-224. doi: 10.1007/s00020-017-2342-5.  Google Scholar

[7]

P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Classics in Applied Mathematics, 40, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898719208.  Google Scholar

[8]

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

[9]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 767-807. doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

[10]

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[11]

S. DuoH. W. van Wyk and Y. Zhang, A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem, J. Comput. Phys., 355 (2018), 233-252.  doi: 10.1016/j.jcp.2017.11.011.  Google Scholar

[12]

S. DuoH. Wang and Y. Zhang, A comparative study on nonlocal diffusion operators related to the fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 231-256.  doi: 10.3934/dcdsb.2018110.  Google Scholar

[13]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/gsm/019.  Google Scholar

[14]

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

[15]

B. Gustafsson, A simple proof of the regularity theorem for the variational inequality of the obstacle problem, Nonlinear Anal., 10 (1986), 1487-1490.  doi: 10.1016/0362-546X(86)90119-7.  Google Scholar

[16]

R. Klages, G. Radons and I. M. Sokolov, Anomalous Transport: Foundations and Applications, Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, 2008. doi: 10.1002/9783527622979.  Google Scholar

[17]

D. Kinderlehrerand and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, 88, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. doi: 10.1137/1.9780898719451.  Google Scholar

[18]

M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator, Fract. Calc. Appl. Anal., 20 (2017), 7-51.  doi: 10.1515/fca-2017-0002.  Google Scholar

[19]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 1, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 181, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[20]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 2, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 182, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65217-2.  Google Scholar

[21]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 3, translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, 183, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65393-3.  Google Scholar

[22]

C. W. K. Lo and J. F. Rodrigues, On a class of fractional obstacle type problems related to the distributional Riesz derivative, preprint, arXiv: 2101.06863. Google Scholar

[23]

H. Lewy and G. Stampacchia, On the regularity of the solution of a variational inequality, Comm. Pure Appl. Math., 22 (1969), 153-188.  doi: 10.1002/cpa.3160220203.  Google Scholar

[24]

W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press. Cambridge, 2000.  Google Scholar

[25]

S. E. Mikhailov, Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains, J. Math. Anal. Appl., 378 (2011), 324-342.  doi: 10.1016/j.jmaa.2010.12.027.  Google Scholar

[26]

R. Musina and A. I. Nazarov, On fractional Laplacians, Comm. Partial Differential Equations, 39 (2014), 1780-1790.  doi: 10.1080/03605302.2013.864304.  Google Scholar

[27]

R. Musina and A. I. Nazarov, On fractional Laplacians - 2, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1667-1673.  doi: 10.1016/j.anihpc.2015.08.001.  Google Scholar

[28]

R. Musina and A. I. Nazarov, Variational inequalities for the spectral fractional Laplacian, Comput. Math. Math. Phys., 57 (2017), 373-386. doi: 10.1134/S0965542517030113.  Google Scholar

[29]

R. Musina, A. I. Nazarov and K. Sreenadh, Variational inequalities for the fractional Laplacian, Potential Anal., 46 (2017), 485-498. doi: 10.1007/s11118-016-9591-9.  Google Scholar

[30]

R. H. NochettoE. Otárola and A. J. Salgado, A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.  doi: 10.1007/s10208-014-9208-x.  Google Scholar

[31]

A. Rüland, Unique continuation for fractional Schrödinger equations with rough potentials, Comm. Partial Differential Equations, 40 (2015), 77-114.  doi: 10.1080/03605302.2014.905594.  Google Scholar

[32]

A. Rüland and J.-N. Wang, On the Fractional Landis Conjecture, J. Funct. Anal., 277 (2019), 3236-3270.  doi: 10.1016/j.jfa.2019.05.026.  Google Scholar

[33]

R. L. Schilling, An introduction to Lévy and Feller processes, in From Lévy-Type Processes to Parabolic SPDEs, by D. Khoshnevisan and R. Schilling (eds. F. Utzet and L. Quer-Sardanyons), 1-126, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2016. doi: 10.1007/978-3-319-34120-0.  Google Scholar

[34]

R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/049.  Google Scholar

[35]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

[36]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126.  doi: 10.4171/RMI/750.  Google Scholar

[37]

M. Warma, The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets, Potential Anal., 42 (2015), 499-547.  doi: 10.1007/s11118-014-9443-4.  Google Scholar

[38]

K. Yosida, Functional Analysis, 6$^{th}$ edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 123. Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/978-3-642-61859-8.  Google Scholar

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