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(Non)local and (non)linear free boundary problems
Fractional Laplacians, perimeters and heat semigroups in Carnot groups
1. | Dip. di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, 40126, Bologna, Italy |
2. | Dip. di Matematica e Informatica, Università di Ferrara, via Machiavelli 30, 44121 Ferrara, Italy |
3. | Dip. di Matematica e Fisica "Ennio De Giorgi", Università del Salento, P.O.B. 193, and INFN, 73100 Lecce, Italy |
4. | Dip. di Matematica, Università di Trento, via Sommarive, 14, 38123 Povo (TN), Italy |
5. | Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, MD 21218, USA |
We define and study the fractional Laplacian and the fractional perimeter of a set in Carnot groups and we compare the perimeter with the asymptotic behaviour of the fractional heat semigroup.
References:
[1] |
L. Ambrosio, M. Miranda Jr and D. Pallara,
Some fine properties of $BV$ functions on Wiener spaces, Anal. Geom. Metr. Spaces, 3 (2015), 212-230.
doi: 10.1515/agms-2015-0013. |
[2] |
L. Angiuli, M. Miranda Jr, D. Pallara and F. Paronetto,
$BV$ functions and parabolic initial boundary value problems on domains, Ann. Mat. Pura Appl.(4), 188 (2009), 297-311.
doi: 10.1007/s10231-008-0076-3. |
[3] |
R. M. Blumenthal and R. K. Getoor,
Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.
doi: 10.1090/S0002-9947-1960-0119247-6. |
[4] |
item BLUbook (MR2363343) A. Bonfiglioli, E. Lanconelli and F. Uguzzoni,
Stratified Lie Groups and Potential Theory for Their Sub-Laplacians,
Springer, 2007. |
[5] |
N. Bouleau and F. Hirsch,
Dirichlet Forms and Analysis on Wiener Spaces, vol 14 of De Gruyter studies in Mathematics, De Gruyter, 1991.
doi: 10.1515/9783110858389. |
[6] |
J. Bourgain, H. Brezis and P. Mironescu,
Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman and A. Sulem (Eds), in: J. L. Menaldi,
E. Rofman and A. Sulem (Eds), Optimal Control and Partial Differential Equations (volume
in Honour of A. Bensoussan’s 60th birthday), IOS Press, (2001), 439-455.
|
[7] |
M. Bramanti, M. Miranda Jr and D. Pallara,
Two characterization of $BV$ functions on Carnot groups via the heat semigroup, Int. Math. Res. Not, 17 (2012), 3845-3876.
doi: 10.1093/imrn/rnr170. |
[8] |
C. Bucur and E. Valdinoci,
Nonlocal Diffusion and Applications, UMI Lectute Notes 20, Springer, 2016.
doi: 10.1007/978-3-319-28739-3. |
[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] |
L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson,
An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhäuser, 2007. |
[11] |
Z.-Q. Chen and T. Kumagai,
Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140 (2008), 277-317.
doi: 10.1007/s00440-007-0070-5. |
[12] |
W. Cygan and T. Grzywny,
Heat content for convolution semigroups, J. Math. Anal. Appl, 446 (2017), 1393-1414.
doi: 10.1016/j.jmaa.2016.09.051. |
[13] |
J. Davila,
On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2001), 519-527.
doi: 10.1007/s005260100135. |
[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] |
F. Ferrari and B. Franchi,
Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458.
doi: 10.1007/s00209-014-1376-5. |
[16] |
G. B. Folland,
Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.
doi: 10.1007/BF02386204. |
[17] |
G. B. Folland and E. Stein,
Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982. |
[18] |
M. Kwaśnicki,
Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal., 20 (2017), 17-51.
doi: 10.1515/fca-2017-0002. |
[19] |
M. Ludwig,
Anisotropic fractional perimeters, J. Diff. Geom., 96 (2014), 77-93.
doi: 10.4310/jdg/1391192693. |
[20] |
M. Ludwig,
Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.
doi: 10.1016/j.aim.2013.10.024. |
[21] |
M. Miranda Jr, D. Pallara, F. Paronetto and M. Preunkert,
Short-time heat flow and functions of bounded variation in $\mathbf{R}^{n}$, Ann. Fac. Sci. Toulouse Math., 16 (2007), 125-145.
doi: 10.5802/afst.1142. |
[22] |
H. M. Nguyen, A. Pinamonti, M. Squassina and E. Vecchi, New characterization of magnetic Sobolev spaces,
arXiv:1703. 09801. |
[23] |
A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula Adv. Calc. Var. (2017).
doi: 10.1515/acv-2017-0019. |
[24] |
A. Pinamonti, M. Squassina and E. Vecchi,
The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.
doi: 10.1016/j.jmaa.2016.12.065. |
[25] |
A. Ponce,
A new approach to Sobolev spaces and connections to $Γ$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.
doi: 10.1007/s00526-003-0195-z. |
[26] |
K. Saka,
Besov spaces and Sobolev spaces on a nilpotent Lie group, Tohoku Math. Journ., 31 (1979), 383-437.
doi: 10.2748/tmj/1178229728. |
[27] |
A. Sánchez-Calle,
Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math., 78 (1984), 143-160.
doi: 10.1007/BF01388721. |
[28] |
item St (MR1232192) E. M. Stein,
Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993. |
[29] |
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. |
[30] |
show all references
References:
[1] |
L. Ambrosio, M. Miranda Jr and D. Pallara,
Some fine properties of $BV$ functions on Wiener spaces, Anal. Geom. Metr. Spaces, 3 (2015), 212-230.
doi: 10.1515/agms-2015-0013. |
[2] |
L. Angiuli, M. Miranda Jr, D. Pallara and F. Paronetto,
$BV$ functions and parabolic initial boundary value problems on domains, Ann. Mat. Pura Appl.(4), 188 (2009), 297-311.
doi: 10.1007/s10231-008-0076-3. |
[3] |
R. M. Blumenthal and R. K. Getoor,
Some theorems on stable processes, Trans. Amer. Math. Soc., 95 (1960), 263-273.
doi: 10.1090/S0002-9947-1960-0119247-6. |
[4] |
item BLUbook (MR2363343) A. Bonfiglioli, E. Lanconelli and F. Uguzzoni,
Stratified Lie Groups and Potential Theory for Their Sub-Laplacians,
Springer, 2007. |
[5] |
N. Bouleau and F. Hirsch,
Dirichlet Forms and Analysis on Wiener Spaces, vol 14 of De Gruyter studies in Mathematics, De Gruyter, 1991.
doi: 10.1515/9783110858389. |
[6] |
J. Bourgain, H. Brezis and P. Mironescu,
Another look at Sobolev spaces, in: J. L. Menaldi, E. Rofman and A. Sulem (Eds), in: J. L. Menaldi,
E. Rofman and A. Sulem (Eds), Optimal Control and Partial Differential Equations (volume
in Honour of A. Bensoussan’s 60th birthday), IOS Press, (2001), 439-455.
|
[7] |
M. Bramanti, M. Miranda Jr and D. Pallara,
Two characterization of $BV$ functions on Carnot groups via the heat semigroup, Int. Math. Res. Not, 17 (2012), 3845-3876.
doi: 10.1093/imrn/rnr170. |
[8] |
C. Bucur and E. Valdinoci,
Nonlocal Diffusion and Applications, UMI Lectute Notes 20, Springer, 2016.
doi: 10.1007/978-3-319-28739-3. |
[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] |
L. Capogna, D. Danielli, S. D. Pauls and J. T. Tyson,
An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Birkhäuser, 2007. |
[11] |
Z.-Q. Chen and T. Kumagai,
Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields, 140 (2008), 277-317.
doi: 10.1007/s00440-007-0070-5. |
[12] |
W. Cygan and T. Grzywny,
Heat content for convolution semigroups, J. Math. Anal. Appl, 446 (2017), 1393-1414.
doi: 10.1016/j.jmaa.2016.09.051. |
[13] |
J. Davila,
On an open question about functions of bounded variation, Calc. Var. Partial Differential Equations, 15 (2001), 519-527.
doi: 10.1007/s005260100135. |
[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] |
F. Ferrari and B. Franchi,
Harnack inequality for fractional sub-Laplacians in Carnot groups, Math. Z., 279 (2015), 435-458.
doi: 10.1007/s00209-014-1376-5. |
[16] |
G. B. Folland,
Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat., 13 (1975), 161-207.
doi: 10.1007/BF02386204. |
[17] |
G. B. Folland and E. Stein,
Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982. |
[18] |
M. Kwaśnicki,
Ten equivalent definitions of the fractional Laplace operator, Frac. Calc. Appl. Anal., 20 (2017), 17-51.
doi: 10.1515/fca-2017-0002. |
[19] |
M. Ludwig,
Anisotropic fractional perimeters, J. Diff. Geom., 96 (2014), 77-93.
doi: 10.4310/jdg/1391192693. |
[20] |
M. Ludwig,
Anisotropic fractional Sobolev norms, Adv. Math., 252 (2014), 150-157.
doi: 10.1016/j.aim.2013.10.024. |
[21] |
M. Miranda Jr, D. Pallara, F. Paronetto and M. Preunkert,
Short-time heat flow and functions of bounded variation in $\mathbf{R}^{n}$, Ann. Fac. Sci. Toulouse Math., 16 (2007), 125-145.
doi: 10.5802/afst.1142. |
[22] |
H. M. Nguyen, A. Pinamonti, M. Squassina and E. Vecchi, New characterization of magnetic Sobolev spaces,
arXiv:1703. 09801. |
[23] |
A. Pinamonti, M. Squassina and E. Vecchi, Magnetic BV functions and the Bourgain-Brezis-Mironescu formula Adv. Calc. Var. (2017).
doi: 10.1515/acv-2017-0019. |
[24] |
A. Pinamonti, M. Squassina and E. Vecchi,
The Maz'ya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., 449 (2017), 1152-1159.
doi: 10.1016/j.jmaa.2016.12.065. |
[25] |
A. Ponce,
A new approach to Sobolev spaces and connections to $Γ$-convergence, Calc. Var. Partial Differential Equations, 19 (2004), 229-255.
doi: 10.1007/s00526-003-0195-z. |
[26] |
K. Saka,
Besov spaces and Sobolev spaces on a nilpotent Lie group, Tohoku Math. Journ., 31 (1979), 383-437.
doi: 10.2748/tmj/1178229728. |
[27] |
A. Sánchez-Calle,
Fundamental solutions and geometry of the sum of squares of vector fields, Invent. Math., 78 (1984), 143-160.
doi: 10.1007/BF01388721. |
[28] |
item St (MR1232192) E. M. Stein,
Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, 1993. |
[29] |
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. |
[30] |
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