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doi: 10.3934/dcds.2022076
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Symmetrization for fractional nonlinear elliptic problems

1. 

Dipartimento di Matematica e Applicazioni "Renato Caccioppoli", Università degli Studi di Napoli Federico Ⅱ, Via Cintia, Monte S. Angelo, I-80126 Napoli, Italy

2. 

Dipartimento di Scienze e Tecnologie, Università degli Studi di Napoli "Parthenope", Centro Direzionale Isola C4, 80143 Napoli, Italy

*Corresponding author: Bruno Volzone

Dedicated to Juan Luis Vázquez, a maestro and friend, for his 75th birthday

Received  February 2021 Revised  May 2022 Early access June 2022

In this note we prove a new symmetrization result, in the form of mass concentration comparison, for solutions of nonlocal nonlinear Dirichlet problems involving fractional $ p $ Laplacians. Some regularity estimates of solutions will be established as a direct application of the main result.

Citation: Vincenzo Ferone, Bruno Volzone. Symmetrization for fractional nonlinear elliptic problems. Discrete and Continuous Dynamical Systems, doi: 10.3934/dcds.2022076
References:
[1]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Am. Math. Soc., 2 (1989), 683-773.  doi: 10.1090/S0894-0347-1989-1002633-4.

[2]

A. AlvinoV. Ferone and G. Trombetti, On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal., 29 (1998), 437-451.  doi: 10.1137/S0036141096302111.

[3]

A. AlvinoG. Trombetti and P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal., Theory Methods Appl., 13 (1989), 185-220.  doi: 10.1016/0362-546X(89)90043-6.

[4]

C. Bandle, Isoperimetric Inequalities and Applications, vol. 7 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass. -London, 1980.

[5]

B. BarriosI. Peral and S. Vita, Some remarks about the summability of nonlocal nonlinear problems, Adv. Nonlinear Anal., 4 (2015), 91-107.  doi: 10.1515/anona-2015-0012.

[6]

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

[7]

M. F. BettaV. Ferone and A. Mercaldo, Regularity for solutions of nonlinear elliptic equations, Bull. Sci. Math., 118 (1994), 539-567. 

[8]

L. BrascoE. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces and Free Boundaries, 16 (2014), 419-458.  doi: 10.4171/IFB/325.

[9]

L. BrascoE. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional $p$-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.

[10]

L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.

[11]

L. BrascoE. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.

[12]

A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561-582.  doi: 10.1016/j.jfa.2005.08.010.

[13]

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.

[14]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differ. Equ., 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6.

[15]

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.

[16]

K. M. Chong and N. M. Rice, Equimeasurable Rearrangements of Functions., Queen's Papers in Pure and Applied Mathematics, 28. Kingston, Ontario, Canada: Queen's University. VI, 1971, 177 pp.

[17]

A. Cianchi and V. Maz'ya, Gradient regularity via rearrangements for $p$-Laplacian type elliptic boundary value problems, J. Eur. Math. Soc. (JEMS), 16 (2014), 571-595.  doi: 10.4171/JEMS/440.

[18]

F. del TesoD. Gómez-Castro and J. L. Vázquez, Three representations of the fractional $p$-Laplacian: Semigroup, extension and Balakrishnan formulas, Fract. Calc. Appl. Anal., 24 (2021), 966-1002.  doi: 10.1515/fca-2021-0042.

[19]

F. del Teso and E. Lindgren, A finite difference method for the variational $p$-Laplacian, J. Sci. Comput., 90 (2022), Paper No. 67, 31 pp. doi: 10.1007/s10915-021-01745-z.

[20]

E. Di NezzaG. 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.

[21]

V. Ferone and B. Messano, A symmetrization result for nonlinear elliptic equations, Rev. Mat. Complut., 17 (2004), 261-276.  doi: 10.5209/rev_REMA.2004.v17.n2.16718.

[22]

V. Ferone and B. Volzone, Symmetrization for fractional elliptic problems: A direct approach, Arch. Ration. Mech. Anal., 239 (2021), 1733-1770.  doi: 10.1007/s00205-020-01601-8.

[23] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 
[24]

R. A. Hunt, On $L(p, q)$ spaces, Enseign. Math. (2), 12 (1966), 249-276. 

[25]

A. IannizzottoS. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.

[26]

S. Kesavan, Symmetrization & Applications, vol. 3 of Series in Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812773937.

[27]

T. KuusiG. Mingione and Y. Sire, Nonlocal equations with measure data, Commun. Math. Phys., 337 (2015), 1317-1368.  doi: 10.1007/s00220-015-2356-2.

[28]

M. Ludwig, Anisotropic fractional perimeters, J. Differ. Geom., 96 (2014), 77-93.  doi: 10.4310/jdg/1391192693.

[29]

W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed., vol. 52, Springer, Berlin, 1966. doi: 10.1007/978-3-662-11761-3.

[30]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke, Math. J., 30 (1963), 129-142.  doi: 10.1215/S0012-7094-63-03015-1.

[31]

Y. SireJ. 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.

[32]

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

[33]

G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4), 120 (1979), 159-184.  doi: 10.1007/BF02411942.

[34]

G. Talenti, Inequalities in rearrangement invariant function spaces, in Nonlinear Analysis, Function Spaces and Applications, Vol. 5 (Prague, 1994), Prometheus, Prague, 1994, 177–230.

[35]

J. L. Vázquez, Symétrisation pour $u_{t}=\Delta \varphi (u)$ et applications, C. R. Acad. Sci. Paris Sér. I Math., 295 (1982), 71-74. 

[36]

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.

show all references

References:
[1]

F. J. Almgren and E. H. Lieb, Symmetric decreasing rearrangement is sometimes continuous, J. Am. Math. Soc., 2 (1989), 683-773.  doi: 10.1090/S0894-0347-1989-1002633-4.

[2]

A. AlvinoV. Ferone and G. Trombetti, On the properties of some nonlinear eigenvalues, SIAM J. Math. Anal., 29 (1998), 437-451.  doi: 10.1137/S0036141096302111.

[3]

A. AlvinoG. Trombetti and P.-L. Lions, On optimization problems with prescribed rearrangements, Nonlinear Anal., Theory Methods Appl., 13 (1989), 185-220.  doi: 10.1016/0362-546X(89)90043-6.

[4]

C. Bandle, Isoperimetric Inequalities and Applications, vol. 7 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, Mass. -London, 1980.

[5]

B. BarriosI. Peral and S. Vita, Some remarks about the summability of nonlocal nonlinear problems, Adv. Nonlinear Anal., 4 (2015), 91-107.  doi: 10.1515/anona-2015-0012.

[6]

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

[7]

M. F. BettaV. Ferone and A. Mercaldo, Regularity for solutions of nonlinear elliptic equations, Bull. Sci. Math., 118 (1994), 539-567. 

[8]

L. BrascoE. Lindgren and E. Parini, The fractional Cheeger problem, Interfaces and Free Boundaries, 16 (2014), 419-458.  doi: 10.4171/IFB/325.

[9]

L. BrascoE. Lindgren and A. Schikorra, Higher Hölder regularity for the fractional $p$-Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782-846.  doi: 10.1016/j.aim.2018.09.009.

[10]

L. Brasco and E. Parini, The second eigenvalue of the fractional $p$-Laplacian, Adv. Calc. Var., 9 (2016), 323-355.  doi: 10.1515/acv-2015-0007.

[11]

L. BrascoE. Parini and M. Squassina, Stability of variational eigenvalues for the fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1813-1845.  doi: 10.3934/dcds.2016.36.1813.

[12]

A. Burchard and H. Hajaiej, Rearrangement inequalities for functionals with monotone integrands, J. Funct. Anal., 233 (2006), 561-582.  doi: 10.1016/j.jfa.2005.08.010.

[13]

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.

[14]

L. Caffarelli and E. Valdinoci, Uniform estimates and limiting arguments for nonlocal minimal surfaces, Calc. Var. Partial Differ. Equ., 41 (2011), 203-240.  doi: 10.1007/s00526-010-0359-6.

[15]

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.

[16]

K. M. Chong and N. M. Rice, Equimeasurable Rearrangements of Functions., Queen's Papers in Pure and Applied Mathematics, 28. Kingston, Ontario, Canada: Queen's University. VI, 1971, 177 pp.

[17]

A. Cianchi and V. Maz'ya, Gradient regularity via rearrangements for $p$-Laplacian type elliptic boundary value problems, J. Eur. Math. Soc. (JEMS), 16 (2014), 571-595.  doi: 10.4171/JEMS/440.

[18]

F. del TesoD. Gómez-Castro and J. L. Vázquez, Three representations of the fractional $p$-Laplacian: Semigroup, extension and Balakrishnan formulas, Fract. Calc. Appl. Anal., 24 (2021), 966-1002.  doi: 10.1515/fca-2021-0042.

[19]

F. del Teso and E. Lindgren, A finite difference method for the variational $p$-Laplacian, J. Sci. Comput., 90 (2022), Paper No. 67, 31 pp. doi: 10.1007/s10915-021-01745-z.

[20]

E. Di NezzaG. 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.

[21]

V. Ferone and B. Messano, A symmetrization result for nonlinear elliptic equations, Rev. Mat. Complut., 17 (2004), 261-276.  doi: 10.5209/rev_REMA.2004.v17.n2.16718.

[22]

V. Ferone and B. Volzone, Symmetrization for fractional elliptic problems: A direct approach, Arch. Ration. Mech. Anal., 239 (2021), 1733-1770.  doi: 10.1007/s00205-020-01601-8.

[23] G. H. HardyJ. E. Littlewood and G. Pólya, Inequalities, Cambridge, at the University Press, 1952. 
[24]

R. A. Hunt, On $L(p, q)$ spaces, Enseign. Math. (2), 12 (1966), 249-276. 

[25]

A. IannizzottoS. Mosconi and M. Squassina, Global Hölder regularity for the fractional $p$-Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353-1392.  doi: 10.4171/RMI/921.

[26]

S. Kesavan, Symmetrization & Applications, vol. 3 of Series in Analysis, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/9789812773937.

[27]

T. KuusiG. Mingione and Y. Sire, Nonlocal equations with measure data, Commun. Math. Phys., 337 (2015), 1317-1368.  doi: 10.1007/s00220-015-2356-2.

[28]

M. Ludwig, Anisotropic fractional perimeters, J. Differ. Geom., 96 (2014), 77-93.  doi: 10.4310/jdg/1391192693.

[29]

W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed., vol. 52, Springer, Berlin, 1966. doi: 10.1007/978-3-662-11761-3.

[30]

R. O'Neil, Convolution operators and $L(p, q)$ spaces, Duke, Math. J., 30 (1963), 129-142.  doi: 10.1215/S0012-7094-63-03015-1.

[31]

Y. SireJ. 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.

[32]

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

[33]

G. Talenti, Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Ann. Mat. Pura Appl. (4), 120 (1979), 159-184.  doi: 10.1007/BF02411942.

[34]

G. Talenti, Inequalities in rearrangement invariant function spaces, in Nonlinear Analysis, Function Spaces and Applications, Vol. 5 (Prague, 1994), Prometheus, Prague, 1994, 177–230.

[35]

J. L. Vázquez, Symétrisation pour $u_{t}=\Delta \varphi (u)$ et applications, C. R. Acad. Sci. Paris Sér. I Math., 295 (1982), 71-74. 

[36]

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.

Figure 1.  From left to right: plot of $ u $ with the choice $ f=|x| $, plot of $ v $ and comparison of mass concentrations of $ p-1 $ powers of $ u^{\#} $ and $ v $
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