On the loss of maximum principles for higher-order fractional Laplacians

Abatangelo, Nicola; Jarohs, Sven; Saldaña, Alberto

doi:10.1090/proc/14165

On the loss of maximum principles for higher-order fractional Laplacians
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by Nicola Abatangelo, Sven Jarohs and Alberto Saldaña PDF

Proc. Amer. Math. Soc. 146 (2018), 4823-4835 Request permission

Abstract:

We study the existence and positivity of solutions to problems involving higher-order fractional Laplacians $(-\Delta )^s$ for any $s>1$. In particular, using a suitable variational framework and the nonlocal properties of these operators, we provide an explicit counterexample to general maximum principles for $s\in (n,n+1)$ with $n\in \mathbb N$ odd, and we mention some particular domains where positivity preserving properties do hold.

References

N. Abatangelo, S. Jarohs, and A. Saldaña, Positive powers of the Laplacian: From hypersingular integrals to boundary value problems, Comm. Pure Appl. Anal., 17 (2008), no. 3, 899-922.
Nicola Abatangelo, Sven Jarohs, and Alberto Saldaña, Green function and Martin kernel for higher-order fractional Laplacians in balls, Nonlinear Anal. 175 (2018), 173–190. MR 3830727, DOI 10.1016/j.na.2018.05.019
N. Abatangelo, S. Jarohs, and A. Saldaña, Integral representation of solutions to higher-order fractional Dirichlet problems on balls, Comm. Cont. Math., to appear.
Gérard Bourdaud and Yves Meyer, Fonctions qui opèrent sur les espaces de Sobolev, J. Funct. Anal. 97 (1991), no. 2, 351–360 (French). MR 1111186, DOI 10.1016/0022-1236(91)90006-Q
Claudia Bucur and Enrico Valdinoci, Nonlocal diffusion and applications, Lecture Notes of the Unione Matematica Italiana, vol. 20, Springer, [Cham]; Unione Matematica Italiana, Bologna, 2016. MR 3469920, DOI 10.1007/978-3-319-28739-3
Xavier Cabré and Yannick Sire, Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré C Anal. Non Linéaire 31 (2014), no. 1, 23–53. MR 3165278, DOI 10.1016/j.anihpc.2013.02.001
Luis Caffarelli and Luis Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245–1260. MR 2354493, DOI 10.1080/03605300600987306
Sun-Yung A. Chang and Paul C. Yang, On uniqueness of solutions of $n$th order differential equations in conformal geometry, Math. Res. Lett. 4 (1997), no. 1, 91–102. MR 1432813, DOI 10.4310/MRL.1997.v4.n1.a9
Serena Dipierro and Hans-Christoph Grunau, Boggio’s formula for fractional polyharmonic Dirichlet problems, Ann. Mat. Pura Appl. (4) 196 (2017), no. 4, 1327–1344. MR 3673669, DOI 10.1007/s10231-016-0618-z
Bartłomiej Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal. 15 (2012), no. 4, 536–555. MR 2974318, DOI 10.2478/s13540-012-0038-8
Mouhamed Moustapha Fall and Tobias Weth, Monotonicity and nonexistence results for some fractional elliptic problems in the half-space, Commun. Contemp. Math. 18 (2016), no. 1, 1550012, 25. MR 3454619, DOI 10.1142/S0219199715500121
Mostafa Fazly and Juncheng Wei, On stable solutions of the fractional Hénon-Lane-Emden equation, Commun. Contemp. Math. 18 (2016), no. 5, 1650005, 24. MR 3523191, DOI 10.1142/S021919971650005X
Filippo Gazzola, Hans-Christoph Grunau, and Guido Sweers, Polyharmonic boundary value problems, Lecture Notes in Mathematics, vol. 1991, Springer-Verlag, Berlin, 2010. Positivity preserving and nonlinear higher order elliptic equations in bounded domains. MR 2667016, DOI 10.1007/978-3-642-12245-3
Gerd Grubb, Fractional Laplacians on domains, a development of Hörmander’s theory of $\mu$-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. MR 3276603, DOI 10.1016/j.aim.2014.09.018
Gerd Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl. 421 (2015), no. 2, 1616–1634. MR 3258341, DOI 10.1016/j.jmaa.2014.07.081
Moritz Kassmann, A new formulation of Harnack’s inequality for nonlocal operators, C. R. Math. Acad. Sci. Paris 349 (2011), no. 11-12, 637–640 (English, with English and French summaries). MR 2817382, DOI 10.1016/j.crma.2011.04.014
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, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 2, 328–344 (Russian); English transl., Math. USSR-Izv. 34 (1990), no. 2, 337–353. MR 998299, DOI 10.1070/IM1990v034n02ABEH000649
Orlando Lopes and Mihai Mariş, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal. 254 (2008), no. 2, 535–592. MR 2376460, DOI 10.1016/j.jfa.2007.10.004
Ali Maalaoui, Luca Martinazzi, and Armin Schikorra, Blow-up behavior of a fractional Adams-Moser-Trudinger-type inequality in odd dimension, Comm. Partial Differential Equations 41 (2016), no. 10, 1593–1618. MR 3555483, DOI 10.1080/03605302.2016.1222544
Roberta Musina and Alexander I. Nazarov, On fractional Laplacians—2, Ann. Inst. H. Poincaré C Anal. Non Linéaire 33 (2016), no. 6, 1667–1673. MR 3569246, DOI 10.1016/j.anihpc.2015.08.001
R. Musina and A.I. Nazarov, A note on truncations in fractional Sobolev spaces, Bull. Math. Sci., to appear.
Giampiero Palatucci and Adriano Pisante, A global compactness type result for Palais-Smale sequences in fractional Sobolev spaces, Nonlinear Anal. 117 (2015), 1–7. MR 3316602, DOI 10.1016/j.na.2014.12.027
L. A. Peletier and W. C. Troy, Spatial patterns, Progress in Nonlinear Differential Equations and their Applications, vol. 45, Birkhäuser Boston, Inc., Boston, MA, 2001. Higher order models in physics and mechanics. MR 1839555, DOI 10.1007/978-1-4612-0135-9
Xavier Ros-Oton and Joaquim Serra, Local integration by parts and Pohozaev identities for higher order fractional Laplacians, Discrete Contin. Dyn. Syst. 35 (2015), no. 5, 2131–2150. MR 3294242, DOI 10.3934/dcds.2015.35.2131
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
Raffaella Servadei and Enrico Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 4, 831–855. MR 3233760, DOI 10.1017/S0308210512001783
Harold S. Shapiro and Max Tegmark, An elementary proof that the biharmonic Green function of an eccentric ellipse changes sign, SIAM Rev. 36 (1994), no. 1, 99–101. MR 1267051, DOI 10.1137/1036005
Luis Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), no. 1, 67–112. MR 2270163, DOI 10.1002/cpa.20153
Guido Sweers, An elementary proof that the triharmonic Green function of an eccentric ellipse changes sign, Arch. Math. (Basel) 107 (2016), no. 1, 59–62. MR 3514727, DOI 10.1007/s00013-016-0909-z