June  2022, 27(6): 2935-2957. doi: 10.3934/dcdsb.2021167

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 Published  June 2022 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 and Continuous Dynamical Systems - B, 2022, 27 (6) : 2935-2957. doi: 10.3934/dcdsb.2021167
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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[4]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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