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The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity
1. | Dipartimento di Ingegneria, Università degli Studi di Napoli "Parthenope", Centro Direzionale Isola C4, Napoli, 80143, Italy |
2. | Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA 50011, USA |
$0<s<1$ |
$\begin{cases} (-Δ+x·\nabla)^su = f,&\hbox{in}~Ω,\\ u = 0,&\hbox{on}~\partialΩ, \end{cases}$ |
$Ω$ |
$\mathbb{R}^n$ |
$n≥q2$ |
$L^p$ |
$L^p(\log L)^α$ |
$f$ |
$u$ |
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
M. Allen, A fractional free boundary problem related to a plasma problem, Comm. Anal. Geom. , (2015), 13pp, arXiv: 1507.06289 |
[3] |
A. Alvino, G. Trombetti, J. I. Diaz and P. L. Lions,
Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math., 49 (1996), 217-236.
doi: 10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-G. |
[4] |
A. Alvino, G. Trombetti and P.-L. Lions,
Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 37-65.
doi: 10.1016/S0294-1449(16)30303-1. |
[5] |
A. Andersson and P. Sjögren, Ornstein-Uhlenbeck Theory in Finite Dimension, Preprint 2012: 12, Matematiska vetenskaper, Göteborg, 2012. |
[6] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116 Second Edition, Cambridge University Press, Cambridge, UK, 2009.
doi: 10.1017/CBO9780511809781. |
[7] |
C. Bandle,
On symmetrizations in parabolic equations, J. Analyse Math., 30 (1976), 98-112.
doi: 10.1007/BF02786706. |
[8] |
C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1988. |
[9] |
M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro,
A comparison result related to Gauss measure, C. R. Math. Acad. Sci. Paris, 334 (2002), 451-456.
doi: 10.1016/S1631-073X(02)02295-1. |
[10] |
M. F. Betta, F. Chiacchio and A. Ferone,
Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems, Z. angew. Math. Phys., 58 (2007), 37-52.
doi: 10.1007/s00033-005-0044-3. |
[11] |
V. I. Bogachev, Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998.
doi: 10.1090/surv/062. |
[12] |
M. Bonforte, Y. Sire and J. L. Vázquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[13] |
C. Borell,
The Brunn-Minkowski inequality in Gauss space, Invent. Math., 30 (1975), 207-216.
doi: 10.1007/BF01425510. |
[14] |
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. |
[15] |
L. A. Caffarelli and Y. Sire, On some pointwise inequalities involving nonlocal operators, in Harmonic Analysis, Partial Differential Equations and Applications, 1-18, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2017. |
[16] |
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. |
[17] |
F. Chiacchio,
Comparison results for linear parabolic equations in unbounded domains via Gaussian symmetrization, Differential Integral Equations, 17 (2004), 241-258.
|
[18] |
K. M. Chong,
Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications, Canad. J. Math., 26 (1974), 1321-1340.
doi: 10.4153/CJM-1974-126-1. |
[19] |
K. M. Chong and N. M. Rice, Equimeasurable Rearrangements of Functions, Queen's University, Kingston, Ont., 1971. Queen's Papers in Pure and Applied Mathematics, No. 28. |
[20] |
G. di Blasio, F. Feo and M. R. Posteraro,
Regularity results for degenerate elliptic equations related to Gauss measure, Math. Inequal. Appl., 10 (2007), 771-797.
doi: 10.7153/mia-10-72. |
[21] |
G. di Blasio and B. Volzone,
Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations, 253 (2012), 2593-2615.
doi: 10.1016/j.jde.2012.07.004. |
[22] |
A. Ehrhard,
Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. (4), 17 (1984), 317-332.
|
[23] |
H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. |
[24] |
F. Feo and M. R. Posteraro,
Logarithmic Sobolev trace inequalities, Asian J. Math., 17 (2013), 569-582.
doi: 10.4310/AJM.2013.v17.n3.a8. |
[25] |
V. Ferone and A. Mercaldo,
A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554.
doi: 10.1016/S0764-4442(98)85005-2. |
[26] |
J. E. Galé, P. J. Miana and P. R. Stinga,
Extension problem for fractional operators: Semigroups and wave equations, J. Evol. Equ., 13 (2013), 343-368.
doi: 10.1007/s00028-013-0182-6. |
[27] |
G. Grubb,
Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.
doi: 10.1002/mana.201500041. |
[28] |
P. Hajłasz, Sobolev mappings, co-area formula and related topics, in Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000,227-254. |
[29] |
Y. Hashimoto,
A remark on the analyticity of the solutions for non-linear elliptic partial differential equations, Tokyo J. Math., 29 (2006), 271-281.
doi: 10.3836/tjm/1170348166. |
[30] |
G. E. Karadzhov and M. Milman,
Extrapolation theory: New results and applications, J. Approx. Theory, 133 (2005), 38-99.
doi: 10.1016/j.jat.2004.12.003. |
[31] |
J. Martín and M. Milman, Fractional Sobolev inequalities: Symmetrization, isoperimetry and interpolation, Astérisque, (2014), ⅹ+127pp. |
[32] |
V. G. Maz'ja,
Weak solutions of the Dirichlet and Neumann problems, Trudy Moskov. Mat. Obšč., 20 (1969), 137-172.
|
[33] |
J. Mossino and J.-M. Rakotoson,
Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 51-73.
|
[34] |
E. V. Nikitin,
Comparison of two definitions of Besov classes on infinite-dimensional spaces, Math. Notes, 95 (2014), 133-135.
doi: 10.1134/S0001434614010143. |
[35] |
M. Novaga, D. Pallara and Y. Sire,
A fractional isoperimetric problem in the Wiener space, J. Anal. Math., 134 (2018), 787-800.
doi: 10.1007/s11854-018-0026-y. |
[36] |
M. Novaga, D. Pallara and Y. Sire,
A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 815-831.
doi: 10.3934/dcdss.2016030. |
[37] |
E. Priola,
On a Dirchlet problem involving an Ornstein-Uhlenbeck operator, Potential Anal., 18 (2003), 251-287.
doi: 10.1023/A:1020933325029. |
[38] |
J. M. Rakotoson and B. Simon,
Relative rearrangement on a finite measure space. Application to the regularity of weighted monotone rearrangement. Ⅰ, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.), 91 (1997), 17-31.
|
[39] |
Y. Sire, J. L. Vázquez and B. Volzone,
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application, Chin. Ann. Math. Ser. B, 38 (2017), 661-686.
doi: 10.1007/s11401-017-1089-2. |
[40] |
R. Song and Z. Vondraček,
Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Related Fields, 125 (2003), 578-592.
doi: 10.1007/s00440-002-0251-1. |
[41] |
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. |
[42] |
P. R. Stinga and B. Volzone,
Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var. Partial Differential Equations, 54 (2015), 1009-1042.
doi: 10.1007/s00526-014-0815-9. |
[43] |
P. R. Stinga and C. Zhang,
Harnack's inequalities for fractional nonlocal equations, Discrete Contin. Dyn. Syst., 33 (2013), 3153-3170.
doi: 10.3934/dcds.2013.33.3153. |
[44] |
G. Talenti,
Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718.
|
[45] |
J. L. Vázquez,
Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations, Adv. Nonlinear Stud., 5 (2005), 87-131.
doi: 10.1515/ans-2005-0107. |
[46] |
J. L. Vázquez and B. Volzone,
Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582.
doi: 10.1016/j.matpur.2013.07.001. |
[47] |
J. L. Vázquez and B. Volzone,
Optimal estimates for fractional fast diffusion equations, J. Math. Pures Appl., 103 (2015), 535-556.
doi: 10.1016/j.matpur.2014.07.002. |
[48] |
B. Volzone,
Symmetrization for fractional Neumann problems, Nonlinear Anal., 147 (2016), 1-25.
doi: 10.1016/j.na.2016.08.029. |
[49] |
H. F. Weinberger, Symmetrization in uniformly elliptic problems, in Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962,424-428. |
show all references
References:
[1] |
R. A. Adams and J. J. F. Fournier, Sobolev Spaces, vol. 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, 2003. |
[2] |
M. Allen, A fractional free boundary problem related to a plasma problem, Comm. Anal. Geom. , (2015), 13pp, arXiv: 1507.06289 |
[3] |
A. Alvino, G. Trombetti, J. I. Diaz and P. L. Lions,
Elliptic equations and Steiner symmetrization, Comm. Pure Appl. Math., 49 (1996), 217-236.
doi: 10.1002/(SICI)1097-0312(199603)49:3<217::AID-CPA1>3.0.CO;2-G. |
[4] |
A. Alvino, G. Trombetti and P.-L. Lions,
Comparison results for elliptic and parabolic equations via Schwarz symmetrization, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 37-65.
doi: 10.1016/S0294-1449(16)30303-1. |
[5] |
A. Andersson and P. Sjögren, Ornstein-Uhlenbeck Theory in Finite Dimension, Preprint 2012: 12, Matematiska vetenskaper, Göteborg, 2012. |
[6] |
D. Applebaum, Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, 116 Second Edition, Cambridge University Press, Cambridge, UK, 2009.
doi: 10.1017/CBO9780511809781. |
[7] |
C. Bandle,
On symmetrizations in parabolic equations, J. Analyse Math., 30 (1976), 98-112.
doi: 10.1007/BF02786706. |
[8] |
C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1988. |
[9] |
M. F. Betta, F. Brock, A. Mercaldo and M. R. Posteraro,
A comparison result related to Gauss measure, C. R. Math. Acad. Sci. Paris, 334 (2002), 451-456.
doi: 10.1016/S1631-073X(02)02295-1. |
[10] |
M. F. Betta, F. Chiacchio and A. Ferone,
Isoperimetric estimates for the first eigenfunction of a class of linear elliptic problems, Z. angew. Math. Phys., 58 (2007), 37-52.
doi: 10.1007/s00033-005-0044-3. |
[11] |
V. I. Bogachev, Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998.
doi: 10.1090/surv/062. |
[12] |
M. Bonforte, Y. Sire and J. L. Vázquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[13] |
C. Borell,
The Brunn-Minkowski inequality in Gauss space, Invent. Math., 30 (1975), 207-216.
doi: 10.1007/BF01425510. |
[14] |
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. |
[15] |
L. A. Caffarelli and Y. Sire, On some pointwise inequalities involving nonlocal operators, in Harmonic Analysis, Partial Differential Equations and Applications, 1-18, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2017. |
[16] |
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. |
[17] |
F. Chiacchio,
Comparison results for linear parabolic equations in unbounded domains via Gaussian symmetrization, Differential Integral Equations, 17 (2004), 241-258.
|
[18] |
K. M. Chong,
Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications, Canad. J. Math., 26 (1974), 1321-1340.
doi: 10.4153/CJM-1974-126-1. |
[19] |
K. M. Chong and N. M. Rice, Equimeasurable Rearrangements of Functions, Queen's University, Kingston, Ont., 1971. Queen's Papers in Pure and Applied Mathematics, No. 28. |
[20] |
G. di Blasio, F. Feo and M. R. Posteraro,
Regularity results for degenerate elliptic equations related to Gauss measure, Math. Inequal. Appl., 10 (2007), 771-797.
doi: 10.7153/mia-10-72. |
[21] |
G. di Blasio and B. Volzone,
Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations, 253 (2012), 2593-2615.
doi: 10.1016/j.jde.2012.07.004. |
[22] |
A. Ehrhard,
Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. (4), 17 (1984), 317-332.
|
[23] |
H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. |
[24] |
F. Feo and M. R. Posteraro,
Logarithmic Sobolev trace inequalities, Asian J. Math., 17 (2013), 569-582.
doi: 10.4310/AJM.2013.v17.n3.a8. |
[25] |
V. Ferone and A. Mercaldo,
A second order derivation formula for functions defined by integrals, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), 549-554.
doi: 10.1016/S0764-4442(98)85005-2. |
[26] |
J. E. Galé, P. J. Miana and P. R. Stinga,
Extension problem for fractional operators: Semigroups and wave equations, J. Evol. Equ., 13 (2013), 343-368.
doi: 10.1007/s00028-013-0182-6. |
[27] |
G. Grubb,
Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.
doi: 10.1002/mana.201500041. |
[28] |
P. Hajłasz, Sobolev mappings, co-area formula and related topics, in Proceedings on Analysis and Geometry (Russian) (Novosibirsk Akademgorodok, 1999), Izdat. Ross. Akad. Nauk Sib. Otd. Inst. Mat., Novosibirsk, 2000,227-254. |
[29] |
Y. Hashimoto,
A remark on the analyticity of the solutions for non-linear elliptic partial differential equations, Tokyo J. Math., 29 (2006), 271-281.
doi: 10.3836/tjm/1170348166. |
[30] |
G. E. Karadzhov and M. Milman,
Extrapolation theory: New results and applications, J. Approx. Theory, 133 (2005), 38-99.
doi: 10.1016/j.jat.2004.12.003. |
[31] |
J. Martín and M. Milman, Fractional Sobolev inequalities: Symmetrization, isoperimetry and interpolation, Astérisque, (2014), ⅹ+127pp. |
[32] |
V. G. Maz'ja,
Weak solutions of the Dirichlet and Neumann problems, Trudy Moskov. Mat. Obšč., 20 (1969), 137-172.
|
[33] |
J. Mossino and J.-M. Rakotoson,
Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 51-73.
|
[34] |
E. V. Nikitin,
Comparison of two definitions of Besov classes on infinite-dimensional spaces, Math. Notes, 95 (2014), 133-135.
doi: 10.1134/S0001434614010143. |
[35] |
M. Novaga, D. Pallara and Y. Sire,
A fractional isoperimetric problem in the Wiener space, J. Anal. Math., 134 (2018), 787-800.
doi: 10.1007/s11854-018-0026-y. |
[36] |
M. Novaga, D. Pallara and Y. Sire,
A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 815-831.
doi: 10.3934/dcdss.2016030. |
[37] |
E. Priola,
On a Dirchlet problem involving an Ornstein-Uhlenbeck operator, Potential Anal., 18 (2003), 251-287.
doi: 10.1023/A:1020933325029. |
[38] |
J. M. Rakotoson and B. Simon,
Relative rearrangement on a finite measure space. Application to the regularity of weighted monotone rearrangement. Ⅰ, Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.), 91 (1997), 17-31.
|
[39] |
Y. Sire, J. L. Vázquez and B. Volzone,
Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application, Chin. Ann. Math. Ser. B, 38 (2017), 661-686.
doi: 10.1007/s11401-017-1089-2. |
[40] |
R. Song and Z. Vondraček,
Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory Related Fields, 125 (2003), 578-592.
doi: 10.1007/s00440-002-0251-1. |
[41] |
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. |
[42] |
P. R. Stinga and B. Volzone,
Fractional semilinear Neumann problems arising from a fractional Keller-Segel model, Calc. Var. Partial Differential Equations, 54 (2015), 1009-1042.
doi: 10.1007/s00526-014-0815-9. |
[43] |
P. R. Stinga and C. Zhang,
Harnack's inequalities for fractional nonlocal equations, Discrete Contin. Dyn. Syst., 33 (2013), 3153-3170.
doi: 10.3934/dcds.2013.33.3153. |
[44] |
G. Talenti,
Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718.
|
[45] |
J. L. Vázquez,
Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations, Adv. Nonlinear Stud., 5 (2005), 87-131.
doi: 10.1515/ans-2005-0107. |
[46] |
J. L. Vázquez and B. Volzone,
Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582.
doi: 10.1016/j.matpur.2013.07.001. |
[47] |
J. L. Vázquez and B. Volzone,
Optimal estimates for fractional fast diffusion equations, J. Math. Pures Appl., 103 (2015), 535-556.
doi: 10.1016/j.matpur.2014.07.002. |
[48] |
B. Volzone,
Symmetrization for fractional Neumann problems, Nonlinear Anal., 147 (2016), 1-25.
doi: 10.1016/j.na.2016.08.029. |
[49] |
H. F. Weinberger, Symmetrization in uniformly elliptic problems, in Studies in Mathematical Analysis and Related Topics, Stanford Univ. Press, Stanford, Calif., 1962,424-428. |
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