Oscillatory phenomena for higher-order fractional Laplacians
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Nicola Abatangelo
Università di Bologna. Dipartimento di Matematica
Sven Jarohs
Goethe-Universität Frankfurt am Main. Institut für Mathematik
Sven Jarohs
Goethe-Universität Frankfurt am Main. Institut für Mathematik
We collect some peculiarities of higher-order fractional Laplacians (−∆)s , s > 1, with special attention to the range s ∈ (1, 2), which show their oscillatory nature. These include the failure of the polarization and P´olya–Szeg˝o inequalities and the explicit example of a domain with sign-changing first eigenfunction. In spite of these fluctuating behaviours, we prove how the Faber–Krahn inequality still holds for any s > 1 in dimension one.
Paraules clau
polarization inequality, Pólya-Szegö inequality, first eigenfunction, positivity-preserving properties, Faber–Krahn inequality
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Abatangelo, Nicola et al. «Oscillatory phenomena for higher-order fractional Laplacians». Publicacions Matemàtiques, 2024, vol.VOL 68, núm. 1, p. 267-86, http://raco.cat/index.php/PublicacionsMatematiques/article/view/422945.
Referències
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N. Abatangelo, S. Jarohs, and A. Saldana , Integral representation of solutions to higherorder fractional Dirichlet problems on balls, Commun. Contemp. Math. 20(8) (2018), 1850002,36 pp. DOI: 10.1142/S0219199718500025
N. Abatangelo, S. Jarohs, and A. Saldana˜ , On the loss of maximum principles for higherorder fractional Laplacians, Proc. Amer. Math. Soc. 146(11) (2018), 4823–4835. DOI: 10.1090/proc/14165
N. Abatangelo, S. Jarohs, and A. Saldana , Positive powers of the Laplacian: from hypersingular integrals to boundary value problems, Commun. Pure Appl. Anal. 17(3) (2018), 899–922.DOI: 10.3934/cpaa.2018045
N. Abatangelo, S. Jarohs, and A. Saldana˜ , Fractional Laplacians on ellipsoids, Math. Eng. 3(5) (2021), Paper no. 038, 34 pp. DOI: 10.3934/mine.2021038
N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in: Contemporary Research in Elliptic PDEs and Related Topics, Springer INdAM Ser. 33, Springer, Cham, 2019, pp. 1–105. DOI: 10.1007/978-3-030-18921-1_1.
M. S. Ashbaugh and R. D. Benguria, On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J. 78(1) (1995), 1–17. DOI: 10.1215/S0012-7094-95-07801-6
A. Baernstein, II, A unified approach to symmetrization, in: Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math. XXXV, Cambridge University Press, Cambridge, 1994, pp. 47–91.
G. Bourdaud and Y. Meyer, Fonctions qui op`erent sur les espaces de Sobolev, J. Funct. Anal. 97(2) (1991), 351–360.
DOI: 10.1016/0022-1236(91)90006-Q
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lect. Notes Unione Mat. Ital. 20, Springer, [Cham]Unione Matematica Italiana, Bologna, 2016. DOI: 10.1007/978-3-319-28739-3
E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012), 521–573. DOI: 10.1016/j.bulsci.2011.12.004
S. Dipierro and H.-C. Grunau, Boggio’s formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl. (4) 196(4) (2017), 1327–1344. DOI: 10.1007/s10231-016-0618-z
R. J. Duffin, On a question of Hadamard concerning super-biharmonic functions, J. Math. Physics 27 (1949), 253–258. DOI: 10.1002/sapm1948271253
R. J. Duffin, The maximum principle and biharmonic functions, J. Math. Anal. Appl. 3(3) (1961), 399–405. DOI: 10.1016/0022-247X(61)90066-X
B. Dyda, A. Kuznetsov, and M. Kwa´snicki, Eigenvalues of the fractional Laplace operator in the unit ball, J. Lond. Math. Soc. (2) 95(2) (2017), 500–518. DOI: 10.1112/jlms.12024
P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1(4) (1951), 485–524. DOI: 10.2140/PJM.1951.1.485
N. Garofalo, Fractional thoughts, in: New Developments in the Analysis of Nonlocal operators, Contemp. Math. 723, American Mathematical Society, Providence, RI, 2019, pp. 1–135.DOI: 10.1090/conm/723/14569
F. Gazzola, H.-C. Grunau, and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Math. 1991, Springer-Verlag, Berlin, 2010. DOI: 10.1007/978-3-642-12245-3
A. Greco and S. Jarohs, Foliated Schwarz symmetry of solutions to a cooperative system of equations involving nonlocal operators, J. Elliptic Parabol. Equ. 8(1) (2022), 383–417. DOI: 10.1007/s41808-022-00155-y
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Reprint of the 1985 original, With a foreword by Susanne C. Brenner, Classics Appl. Math. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. DOI: 10.1137/1.9781611972030
J. Hadamard, Mémoire sur le problème d’analyse relatif à l’´equilibre des plaques elastiques encastrées, Mem. Sav. etrang. 33 (1908), 1–128. Reprinted in: Mémoire sur le problème d’analyse relatif à l’´equilibre des plaques elastiques encastrées, in: Œuvres de Jacques Hadamard. Tomes
I, II, III, IV, Editions du Centre National de la Recherche Scientifique, Paris, 1968, pp. 515–641.
S. Jarohs, Symmetry of solutions to nonlocal nonlinear boundary value problems in radial sets, NoDEA Nonlinear Differential Equations Appl. 23(3) (2016), Art. 32, 22 pp. DOI: 10.1007/s00030-016-0386-x
S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl. (4) 195(1) (2016), 273–291. DOI: 10.1007/s10231-014-0462-y
V. A. Kozlov, V. A. Kondrat’ev, and V. G. Maz’ya, On sign variability and the absence of “strong” zeros of solutions of elliptic equations (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53(2) (1989), 328–344; translation in: Math. USSR-Izv. 34(2) (1990), 337–353. DOI: 10.1070/IM1990v034n02ABEH000649
O. Lopes and M. Maris¸, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal. 254(2) (2008), 535–592. DOI: 10.1016/j.jfa.2007.10.004
R. Musina and A. I. Nazarov, A note on truncations in fractional Sobolev spaces, Bull. Math. Sci. 9(1) (2019), 1950001, 7 pp. DOI: 10.1142/S1664360719500012
N. S. Nadirashvili, Rayleigh’s conjecture on the principal frequency of the clamped plate, Arch. Rational Mech. Anal. 129(1) (1995), 1–10. DOI: 10.1007/BF00375124
M. Nakai and L. Sario, Green’s function of the clamped punctured disk, J. Austral. Math. Soc. Ser. B 20(2) (1977), 175–181. DOI: 10.1017/S0334270000001557
Y. J. Park, Logarithmic Sobolev trace inequality, Proc. Amer. Math. Soc. 132(7) (2004), 2075–2083. DOI: 10.1090/S0002-9939-03-07329-5
X. Ros-Oton and J. Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst. 35(5) (2015), 2131–2150. DOI: 10.3934/dcds.2015.35.2131
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Edited and with a foreword by S. M.Nikolskii, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon, 1993.
J. B. Seif, On the Green’s function for the biharmonic equation in an infinite wedge, Trans. Amer. Math. Soc. 182 (1973), 241–260. DOI: 10.2307/1996533
H. S. Shapiro and M. Tegmark, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36(1) (1994), 99–101. DOI: 10.1137/1036005
G. Sweers, An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign, Arch. Math. (Basel) 107(1) (2016), 59–62. DOI: 10.1007/s00013-016-0909-z
G. Sweers, Correction to: An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign, Arch. Math. (Basel) 112(2) (2019), 223–224. DOI: 10.1007/s00013-018-1274-x
G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4) (1976), 697–718.
G. Talenti, On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl. (4) 129 (1981), 265–280. DOI: 10.1007/BF01762146
H. Triebel, Interpolation Theory, Function Spaces, Differential operators, North-Holland Math. Library 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
J. van Schaftingen, Universal approximation of symmetrizations by polarizations, Proc. Amer. Math. Soc. 134(1) (2006), 177–186.
DOI: 10.1090/S0002-9939-05-08325-5
J. van Schaftingen and M. Willem, Set transformations, symmetrizations and isoperimetric inequalities, in: Nonlinear Analysis and Applications to Physical Sciences, Springer-Verlag Italia, Milan, 2004, pp. 135–152.
T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver. 112(3) (2010), 119–158. DOI: 10.1365/s13291-010-0005-4
N. Abatangelo, S. Jarohs, and A. Saldana , Integral representation of solutions to higherorder fractional Dirichlet problems on balls, Commun. Contemp. Math. 20(8) (2018), 1850002,36 pp. DOI: 10.1142/S0219199718500025
N. Abatangelo, S. Jarohs, and A. Saldana˜ , On the loss of maximum principles for higherorder fractional Laplacians, Proc. Amer. Math. Soc. 146(11) (2018), 4823–4835. DOI: 10.1090/proc/14165
N. Abatangelo, S. Jarohs, and A. Saldana , Positive powers of the Laplacian: from hypersingular integrals to boundary value problems, Commun. Pure Appl. Anal. 17(3) (2018), 899–922.DOI: 10.3934/cpaa.2018045
N. Abatangelo, S. Jarohs, and A. Saldana˜ , Fractional Laplacians on ellipsoids, Math. Eng. 3(5) (2021), Paper no. 038, 34 pp. DOI: 10.3934/mine.2021038
N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, in: Contemporary Research in Elliptic PDEs and Related Topics, Springer INdAM Ser. 33, Springer, Cham, 2019, pp. 1–105. DOI: 10.1007/978-3-030-18921-1_1.
M. S. Ashbaugh and R. D. Benguria, On Rayleigh’s conjecture for the clamped plate and its generalization to three dimensions, Duke Math. J. 78(1) (1995), 1–17. DOI: 10.1215/S0012-7094-95-07801-6
A. Baernstein, II, A unified approach to symmetrization, in: Partial Differential Equations of Elliptic Type (Cortona, 1992), Sympos. Math. XXXV, Cambridge University Press, Cambridge, 1994, pp. 47–91.
G. Bourdaud and Y. Meyer, Fonctions qui op`erent sur les espaces de Sobolev, J. Funct. Anal. 97(2) (1991), 351–360.
DOI: 10.1016/0022-1236(91)90006-Q
C. Bucur and E. Valdinoci, Nonlocal Diffusion and Applications, Lect. Notes Unione Mat. Ital. 20, Springer, [Cham]Unione Matematica Italiana, Bologna, 2016. DOI: 10.1007/978-3-319-28739-3
E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136(5) (2012), 521–573. DOI: 10.1016/j.bulsci.2011.12.004
S. Dipierro and H.-C. Grunau, Boggio’s formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl. (4) 196(4) (2017), 1327–1344. DOI: 10.1007/s10231-016-0618-z
R. J. Duffin, On a question of Hadamard concerning super-biharmonic functions, J. Math. Physics 27 (1949), 253–258. DOI: 10.1002/sapm1948271253
R. J. Duffin, The maximum principle and biharmonic functions, J. Math. Anal. Appl. 3(3) (1961), 399–405. DOI: 10.1016/0022-247X(61)90066-X
B. Dyda, A. Kuznetsov, and M. Kwa´snicki, Eigenvalues of the fractional Laplace operator in the unit ball, J. Lond. Math. Soc. (2) 95(2) (2017), 500–518. DOI: 10.1112/jlms.12024
P. R. Garabedian, A partial differential equation arising in conformal mapping, Pacific J. Math. 1(4) (1951), 485–524. DOI: 10.2140/PJM.1951.1.485
N. Garofalo, Fractional thoughts, in: New Developments in the Analysis of Nonlocal operators, Contemp. Math. 723, American Mathematical Society, Providence, RI, 2019, pp. 1–135.DOI: 10.1090/conm/723/14569
F. Gazzola, H.-C. Grunau, and G. Sweers, Polyharmonic Boundary Value Problems. Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Domains, Lecture Notes in Math. 1991, Springer-Verlag, Berlin, 2010. DOI: 10.1007/978-3-642-12245-3
A. Greco and S. Jarohs, Foliated Schwarz symmetry of solutions to a cooperative system of equations involving nonlocal operators, J. Elliptic Parabol. Equ. 8(1) (2022), 383–417. DOI: 10.1007/s41808-022-00155-y
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Reprint of the 1985 original, With a foreword by Susanne C. Brenner, Classics Appl. Math. 69, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011. DOI: 10.1137/1.9781611972030
J. Hadamard, Mémoire sur le problème d’analyse relatif à l’´equilibre des plaques elastiques encastrées, Mem. Sav. etrang. 33 (1908), 1–128. Reprinted in: Mémoire sur le problème d’analyse relatif à l’´equilibre des plaques elastiques encastrées, in: Œuvres de Jacques Hadamard. Tomes
I, II, III, IV, Editions du Centre National de la Recherche Scientifique, Paris, 1968, pp. 515–641.
S. Jarohs, Symmetry of solutions to nonlocal nonlinear boundary value problems in radial sets, NoDEA Nonlinear Differential Equations Appl. 23(3) (2016), Art. 32, 22 pp. DOI: 10.1007/s00030-016-0386-x
S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl. (4) 195(1) (2016), 273–291. DOI: 10.1007/s10231-014-0462-y
V. A. Kozlov, V. A. Kondrat’ev, and V. G. Maz’ya, On sign variability and the absence of “strong” zeros of solutions of elliptic equations (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 53(2) (1989), 328–344; translation in: Math. USSR-Izv. 34(2) (1990), 337–353. DOI: 10.1070/IM1990v034n02ABEH000649
O. Lopes and M. Maris¸, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal. 254(2) (2008), 535–592. DOI: 10.1016/j.jfa.2007.10.004
R. Musina and A. I. Nazarov, A note on truncations in fractional Sobolev spaces, Bull. Math. Sci. 9(1) (2019), 1950001, 7 pp. DOI: 10.1142/S1664360719500012
N. S. Nadirashvili, Rayleigh’s conjecture on the principal frequency of the clamped plate, Arch. Rational Mech. Anal. 129(1) (1995), 1–10. DOI: 10.1007/BF00375124
M. Nakai and L. Sario, Green’s function of the clamped punctured disk, J. Austral. Math. Soc. Ser. B 20(2) (1977), 175–181. DOI: 10.1017/S0334270000001557
Y. J. Park, Logarithmic Sobolev trace inequality, Proc. Amer. Math. Soc. 132(7) (2004), 2075–2083. DOI: 10.1090/S0002-9939-03-07329-5
X. Ros-Oton and J. Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst. 35(5) (2015), 2131–2150. DOI: 10.3934/dcds.2015.35.2131
S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Edited and with a foreword by S. M.Nikolskii, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon, 1993.
J. B. Seif, On the Green’s function for the biharmonic equation in an infinite wedge, Trans. Amer. Math. Soc. 182 (1973), 241–260. DOI: 10.2307/1996533
H. S. Shapiro and M. Tegmark, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36(1) (1994), 99–101. DOI: 10.1137/1036005
G. Sweers, An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign, Arch. Math. (Basel) 107(1) (2016), 59–62. DOI: 10.1007/s00013-016-0909-z
G. Sweers, Correction to: An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign, Arch. Math. (Basel) 112(2) (2019), 223–224. DOI: 10.1007/s00013-018-1274-x
G. Talenti, Elliptic equations and rearrangements, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 3(4) (1976), 697–718.
G. Talenti, On the first eigenvalue of the clamped plate, Ann. Mat. Pura Appl. (4) 129 (1981), 265–280. DOI: 10.1007/BF01762146
H. Triebel, Interpolation Theory, Function Spaces, Differential operators, North-Holland Math. Library 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
J. van Schaftingen, Universal approximation of symmetrizations by polarizations, Proc. Amer. Math. Soc. 134(1) (2006), 177–186.
DOI: 10.1090/S0002-9939-05-08325-5
J. van Schaftingen and M. Willem, Set transformations, symmetrizations and isoperimetric inequalities, in: Nonlinear Analysis and Applications to Physical Sciences, Springer-Verlag Italia, Milan, 2004, pp. 135–152.
T. Weth, Symmetry of solutions to variational problems for nonlinear elliptic equations via reflection methods, Jahresber. Dtsch. Math.-Ver. 112(3) (2010), 119–158. DOI: 10.1365/s13291-010-0005-4