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September  2015, 14(5): 2043-2067. doi: 10.3934/cpaa.2015.14.2043

A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains

Mahamadi Warma 1,

1. 

University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377

Received  January 2015 Revised  April 2015 Published  June 2015

Let $\Omega\subset R^N$ be a bounded open set with Lipschitz continuous boundary $\partial \Omega$. We define a fractional Dirichlet-to-Neumann operator and prove that it generates a strongly continuous analytic and compact semigroup on $L^2(\partial \Omega)$ which can also be ultracontractive. We also use the fractional Dirichlet-to-Neumann operator to compare the eigenvalues of a realization in $L^2(\Omega)$ of the fractional Laplace operator with Dirichlet boundary condition and the regional fractional Laplacian with the fractional Neumann boundary conditions.
Keywords: Fractional Laplacian, semigroup., fractional Dirichlet-to-Neumann operator, form methods.
Mathematics Subject Classification: 35R11, 35P15, 47D0.
Citation: Mahamadi Warma. A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2043-2067. doi: 10.3934/cpaa.2015.14.2043
References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.  Google Scholar

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Second edition. Monographs in Mathematics 96, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.  Google Scholar

[3]

W. Arendt and R. Mazzeo, Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212. doi: 10.3934/cpaa.2012.11.2201.  Google Scholar

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W. Arendt and A. F. M. ter Elst, The Dirichlet-to-Neumann operator on rough domains, J. Differential Equations, 251 (2011), 2100-2124. doi: 10.1016/j.jde.2011.06.017.  Google Scholar

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W. Arendt, A. F. M. ter Elst, J. B. Kennedy and M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786. doi: 10.1016/j.jfa.2013.09.012.  Google Scholar

[6]

K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.  Google Scholar

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H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

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L. Caffarelli, J-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.  Google Scholar

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L. 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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D. Danielli, N. Garofalo and D-M. Nhieu, Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces, Mem. Amer. Math. Soc., 182 (2006). doi: 10.1090/memo/0857.  Google Scholar

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D. Daners, Dirichlet problems on varying domains, J. Differential Equations, 188 (2003), 591-624. doi: 10.1016/S0022-0396(02)00105-5.  Google Scholar

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D. Daners, Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator, Positivity, 18 (2014), 235-256. doi: 10.1007/s11117-013-0243-7.  Google Scholar

[13]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

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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

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Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540. doi: 10.1142/S0218202512500546.  Google Scholar

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L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal., 116 (1991), 153-160. doi: 10.1007/BF00375590.  Google Scholar

[17]

H. Gimperlein and G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ., 14 (2014), 49-83. doi: 10.1007/s00028-013-0206-2.  Google Scholar

[18]

Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.  Google Scholar

[19]

Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.  Google Scholar

[20]

W. Hoh and J. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Funct. Anal., 137 (1996), 19-48. doi: 10.1006/jfan.1996.0039.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, Springer Berlin, 1966. Google Scholar

[22]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[23]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[24]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar

[25]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.  Google Scholar

[26]

A. F. M. ter Elst and E. M. Ouhabaz, Analysis of the heat kernel of the Dirichlet-to-Neumann operator, J. Funct. Anal., 267 (2014), 4066-4109. doi: 10.1016/j.jfa.2014.09.001.  Google Scholar

[27]

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

show all references

References:
[1]

D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften 314, Springer-Verlag, Berlin, 1996. doi: 10.1007/978-3-662-03282-4.  Google Scholar

[2]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems, Second edition. Monographs in Mathematics 96, Birkhäuser/Springer Basel AG, Basel, 2011. doi: 10.1007/978-3-0348-0087-7.  Google Scholar

[3]

W. Arendt and R. Mazzeo, Friedlander's eigenvalue inequalities and the Dirichlet-to-Neumann semigroup, Commun. Pure Appl. Anal., 11 (2012), 2201-2212. doi: 10.3934/cpaa.2012.11.2201.  Google Scholar

[4]

W. Arendt and A. F. M. ter Elst, The Dirichlet-to-Neumann operator on rough domains, J. Differential Equations, 251 (2011), 2100-2124. doi: 10.1016/j.jde.2011.06.017.  Google Scholar

[5]

W. Arendt, A. F. M. ter Elst, J. B. Kennedy and M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness, J. Funct. Anal., 266 (2014), 1757-1786. doi: 10.1016/j.jfa.2013.09.012.  Google Scholar

[6]

K. Bogdan, K. Burdzy and Z. Q. Chen, Censored stable processes, Probab. Theory Related Fields, 127 (2003), 89-152. doi: 10.1007/s00440-003-0275-1.  Google Scholar

[7]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011.  Google Scholar

[8]

L. Caffarelli, J-M. Roquejoffre and Y. Sire, Variational problems for free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.  Google Scholar

[9]

L. 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

[10]

D. Danielli, N. Garofalo and D-M. Nhieu, Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces, Mem. Amer. Math. Soc., 182 (2006). doi: 10.1090/memo/0857.  Google Scholar

[11]

D. Daners, Dirichlet problems on varying domains, J. Differential Equations, 188 (2003), 591-624. doi: 10.1016/S0022-0396(02)00105-5.  Google Scholar

[12]

D. Daners, Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator, Positivity, 18 (2014), 235-256. doi: 10.1007/s11117-013-0243-7.  Google Scholar

[13]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989. doi: 10.1017/CBO9780511566158.  Google Scholar

[14]

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

[15]

Q. Du, M. Gunzburger, R. B. Lehoucq and K. Zhou, A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws, Math. Models Methods Appl. Sci., 23 (2013), 493-540. doi: 10.1142/S0218202512500546.  Google Scholar

[16]

L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal., 116 (1991), 153-160. doi: 10.1007/BF00375590.  Google Scholar

[17]

H. Gimperlein and G. Grubb, Heat kernel estimates for pseudodifferential operators, fractional Laplacians and Dirichlet-to-Neumann operators, J. Evol. Equ., 14 (2014), 49-83. doi: 10.1007/s00028-013-0206-2.  Google Scholar

[18]

Q. Y. Guan, Integration by parts formula for regional fractional Laplacian, Comm. Math. Phys., 266 (2006), 289-329. doi: 10.1007/s00220-006-0054-9.  Google Scholar

[19]

Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424. doi: 10.1142/S021949370500150X.  Google Scholar

[20]

W. Hoh and J. Jacob, On the Dirichlet problem for pseudodifferential operators generating Feller semigroups, J. Funct. Anal., 137 (1996), 19-48. doi: 10.1006/jfan.1996.0039.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, Springer Berlin, 1966. Google Scholar

[22]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005.  Google Scholar

[23]

X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.  Google Scholar

[24]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar

[25]

R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 831-855. doi: 10.1017/S0308210512001783.  Google Scholar

[26]

A. F. M. ter Elst and E. M. Ouhabaz, Analysis of the heat kernel of the Dirichlet-to-Neumann operator, J. Funct. Anal., 267 (2014), 4066-4109. doi: 10.1016/j.jfa.2014.09.001.  Google Scholar

[27]

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

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