July  2018, 38(7): 3269-3298. doi: 10.3934/dcds.2018142

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

* Corresponding author: Bruno Volzone

Received  June 2017 Revised  February 2018 Published  April 2018

For
$0<s<1$
, we consider the Dirichlet problem for the fractional nonlocal Ornstein-Uhlenbeck equation
$\begin{cases} (-Δ+x·\nabla)^su = f,&\hbox{in}~Ω,\\ u = 0,&\hbox{on}~\partialΩ, \end{cases}$
where
$Ω$
is a possibly unbounded open subset of
$\mathbb{R}^n$
,
$n≥2$
. The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel
$L^p$
and
$L^p(\log L)^α$
regularity estimates in terms of the datum
$f$
are obtained by comparing
$u$
with half-space solutions.
Citation: Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete and Continuous Dynamical Systems, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
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. AlvinoG. TrombettiJ. 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. AlvinoG. 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. BettaF. BrockA. 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. BettaF. 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. BonforteY. 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 BlasioF. 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. GaléP. 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. NovagaD. 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. NovagaD. 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. SireJ. 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. AlvinoG. TrombettiJ. 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. AlvinoG. 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. BettaF. BrockA. 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. BettaF. 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. BonforteY. 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 BlasioF. 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. GaléP. 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. NovagaD. 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. NovagaD. 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. SireJ. 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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