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 & Continuous Dynamical Systems - A, 2018, 38 (7) : 3269-3298. doi: 10.3934/dcds.2018142
References:
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[2]

M. Allen, A fractional free boundary problem related to a plasma problem, Comm. Anal. Geom., (2015), 13pp, arXiv: 1507.06289 Google Scholar

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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.  Google Scholar

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A. Andersson and P. Sjögren, Ornstein-Uhlenbeck Theory in Finite Dimension, Preprint 2012: 12, Matematiska vetenskaper, Göteborg, 2012. Google Scholar

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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.  Google Scholar

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C. Bandle, On symmetrizations in parabolic equations, J. Analyse Math., 30 (1976), 98-112.  doi: 10.1007/BF02786706.  Google Scholar

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C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math., 30 (1975), 207-216.  doi: 10.1007/BF01425510.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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F. Chiacchio, Comparison results for linear parabolic equations in unbounded domains via Gaussian symmetrization, Differential Integral Equations, 17 (2004), 241-258.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Ehrhard, Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. (4), 17 (1984), 317-332.   Google Scholar

[23]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.  doi: 10.1002/mana.201500041.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Martín and M. Milman, Fractional Sobolev inequalities: Symmetrization, isoperimetry and interpolation, Astérisque, (2014), ⅹ+127pp.  Google Scholar

[32]

V. G. Maz'ja, Weak solutions of the Dirichlet and Neumann problems, Trudy Moskov. Mat. Obšč., 20 (1969), 137-172.   Google Scholar

[33]

J. Mossino and J.-M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 51-73.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

E. Priola, On a Dirchlet problem involving an Ornstein-Uhlenbeck operator, Potential Anal., 18 (2003), 251-287.  doi: 10.1023/A:1020933325029.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

B. Volzone, Symmetrization for fractional Neumann problems, Nonlinear Anal., 147 (2016), 1-25.  doi: 10.1016/j.na.2016.08.029.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. Allen, A fractional free boundary problem related to a plasma problem, Comm. Anal. Geom., (2015), 13pp, arXiv: 1507.06289 Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

A. Andersson and P. Sjögren, Ornstein-Uhlenbeck Theory in Finite Dimension, Preprint 2012: 12, Matematiska vetenskaper, Göteborg, 2012. Google Scholar

[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.  Google Scholar

[7]

C. Bandle, On symmetrizations in parabolic equations, J. Analyse Math., 30 (1976), 98-112.  doi: 10.1007/BF02786706.  Google Scholar

[8]

C. Bennett and R. Sharpley, Interpolation of Operators, vol. 129 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1988.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

V. I. Bogachev, Gaussian Measures, vol. 62 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1998. doi: 10.1090/surv/062.  Google Scholar

[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.  Google Scholar

[13]

C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math., 30 (1975), 207-216.  doi: 10.1007/BF01425510.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F. Chiacchio, Comparison results for linear parabolic equations in unbounded domains via Gaussian symmetrization, Differential Integral Equations, 17 (2004), 241-258.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Ehrhard, Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. (4), 17 (1984), 317-332.   Google Scholar

[23]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Grubb, Regularity of spectral fractional Dirichlet and Neumann problems, Math. Nachr., 289 (2016), 831-844.  doi: 10.1002/mana.201500041.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

J. Martín and M. Milman, Fractional Sobolev inequalities: Symmetrization, isoperimetry and interpolation, Astérisque, (2014), ⅹ+127pp.  Google Scholar

[32]

V. G. Maz'ja, Weak solutions of the Dirichlet and Neumann problems, Trudy Moskov. Mat. Obšč., 20 (1969), 137-172.   Google Scholar

[33]

J. Mossino and J.-M. Rakotoson, Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 51-73.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

E. Priola, On a Dirchlet problem involving an Ornstein-Uhlenbeck operator, Potential Anal., 18 (2003), 251-287.  doi: 10.1023/A:1020933325029.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 3 (1976), 697-718.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[48]

B. Volzone, Symmetrization for fractional Neumann problems, Nonlinear Anal., 147 (2016), 1-25.  doi: 10.1016/j.na.2016.08.029.  Google Scholar

[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.  Google Scholar

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