The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations

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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

Pu-Zhao Kow ^1,, and Masato Kimura ^2,

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.

Keywords: Lewy-Stampacchia, obstacle problem, fractional Laplacian, nonlocal operators, anomalous unidirectional diffusion.

Mathematics Subject Classification: 35R11, 35K61, 35K86.

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:

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[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. Duo, H. 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. Duo, H. 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. Nochetto, E. 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. Duo, H. 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. Duo, H. 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. Nochetto, E. 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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