The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity

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

The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity

Filomena Feo ^1, , Pablo Raúl Stinga ^2, and Bruno Volzone ^1,,

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.

Keywords: Fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization, extension problem, regularity, method of semigroups.

Mathematics Subject Classification: Primary: 35R11, 35B65, 35A01; Secondary: 28C20, 35K08, 46E35, 60J35.

Citation: Filomena Feo, Pablo Raúl Stinga, Bruno Volzone. The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity. Discrete & Continuous Dynamical Systems - A, 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. 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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