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A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains
1. | University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, P.O. Box 70377, San Juan PR 00936-8377 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[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. |
[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. |
[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. |
[11] |
D. Daners, Dirichlet problems on varying domains, J. Differential Equations, 188 (2003), 591-624.
doi: 10.1016/S0022-0396(02)00105-5. |
[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. |
[13] |
E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511566158. |
[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. |
[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. |
[16] |
L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal., 116 (1991), 153-160.
doi: 10.1007/BF00375590. |
[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. |
[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. |
[19] |
Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.
doi: 10.1142/S021949370500150X. |
[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. |
[21] |
T. Kato, Perturbation Theory for Linear Operators, Springer Berlin, 1966. |
[22] |
E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005. |
[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. |
[24] |
R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext. Springer, New York, 2011. |
[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. |
[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. |
[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. |
[11] |
D. Daners, Dirichlet problems on varying domains, J. Differential Equations, 188 (2003), 591-624.
doi: 10.1016/S0022-0396(02)00105-5. |
[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. |
[13] |
E. B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
doi: 10.1017/CBO9780511566158. |
[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. |
[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. |
[16] |
L. Friedlander, Some inequalities between Dirichlet and Neumann eigenvalues, Arch. Rational Mech. Anal., 116 (1991), 153-160.
doi: 10.1007/BF00375590. |
[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. |
[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. |
[19] |
Q. Y. Guan and Z. M. Ma, Boundary problems for fractional Laplacians, Stoch. Dyn., 5 (2005), 385-424.
doi: 10.1142/S021949370500150X. |
[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. |
[21] |
T. Kato, Perturbation Theory for Linear Operators, Springer Berlin, 1966. |
[22] |
E. M. Ouhabaz, Analysis of Heat Equations on Domains, London Mathematical Society Monographs Series 31, Princeton University Press, Princeton, NJ, 2005. |
[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. |
[24] |
R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137. |
[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. |
[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. |
[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. |
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