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

Pu-Zhao Kow; Masato Kimura

doi:10.3934/dcdsb.2021167

Previous Article
Modelling conflicting individual preference: Target sequences and graph realization
DCDS-B Home
This Issue
Next Article
Concentration phenomenon of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with spontaneous infection

doi: 10.3934/dcdsb.2021167

Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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 & Continuous Dynamical Systems - B, 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. 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. 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. Google Scholar
[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. 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. 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. 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. 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. Google Scholar
[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. 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. 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. 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

[1]	Siwei Duo, Hong Wang, Yanzhi Zhang. A comparative study on nonlocal diffusion operators related to the fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 231-256. doi: 10.3934/dcdsb.2018110
[2]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[3]	Dinh-Ke Tran, Tran-Phuong-Thuy Lam. Nonlocal final value problem governed by semilinear anomalous diffusion equations. Evolution Equations & Control Theory, 2020, 9 (3) : 891-914. doi: 10.3934/eect.2020038
[4]	José-Francisco Rodrigues, João Lita da Silva. On a unilateral reaction-diffusion system and a nonlocal evolution obstacle problem. Communications on Pure & Applied Analysis, 2004, 3 (1) : 85-95. doi: 10.3934/cpaa.2004.3.85
[5]	Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857
[6]	Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi. On a nonlocal problem involving the fractional $ p(x,.) $-Laplacian satisfying Cerami condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3479-3495. doi: 10.3934/dcdss.2020425
[7]	Gloria Paoli, Gianpaolo Piscitelli, Rossanno Sannipoli. A stability result for the Steklov Laplacian Eigenvalue Problem with a spherical obstacle. Communications on Pure & Applied Analysis, 2021, 20 (1) : 145-158. doi: 10.3934/cpaa.2020261
[8]	Mikko Kemppainen, Peter Sjögren, José Luis Torrea. Wave extension problem for the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 4905-4929. doi: 10.3934/dcds.2015.35.4905
[9]	Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213
[10]	Yuanwei Qi. Anomalous exponents and RG for nonlinear diffusion equations. Conference Publications, 2005, 2005 (Special) : 738-745. doi: 10.3934/proc.2005.2005.738
[11]	Stephen Thompson, Thomas I. Seidman. Approximation of a semigroup model of anomalous diffusion in a bounded set. Evolution Equations & Control Theory, 2013, 2 (1) : 173-192. doi: 10.3934/eect.2013.2.173
[12]	Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160
[13]	Xudong Shang, Jihui Zhang, Yang Yang. Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Communications on Pure & Applied Analysis, 2014, 13 (2) : 567-584. doi: 10.3934/cpaa.2014.13.567
[14]	Luis Caffarelli, Antoine Mellet. Random homogenization of fractional obstacle problems. Networks & Heterogeneous Media, 2008, 3 (3) : 523-554. doi: 10.3934/nhm.2008.3.523
[15]	Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure & Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177
[16]	Li Li. An inverse problem for a fractional diffusion equation with fractional power type nonlinearities. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021064
[17]	Seung-Yeal Ha, Jinwook Jung, Peter Kuchling. Emergence of anomalous flocking in the fractional Cucker-Smale model. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5465-5489. doi: 10.3934/dcds.2019223
[18]	Nikolaos Roidos, Yuanzhen Shao. Functional inequalities involving nonlocal operators on complete Riemannian manifolds and their applications to the fractional porous medium equation. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021026
[19]	Hélène Hivert. Numerical schemes for kinetic equation with diffusion limit and anomalous time scale. Kinetic & Related Models, 2018, 11 (2) : 409-439. doi: 10.3934/krm.2018019
[20]	Gernot Holler, Karl Kunisch. Learning nonlocal regularization operators. Mathematical Control & Related Fields, 2022, 12 (1) : 81-114. doi: 10.3934/mcrf.2021003

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors