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One-parameter solutions of the Euler-Arnold equation on the contactomorphism group
Local integration by parts and Pohozaev identities for higher order fractional Laplacians
1. | The University of Texas at Austin, Department of Mathematics, 2515 Speedway, Austin, TX 78751, United States |
2. | Universitat Politècnica de Catalunya, Departament de Matemàtica Aplicada I, Avda. Diagonal 647, 08028 Barcelona, Spain |
As an immediate consequence of these results, we obtain a unique continuation property for the eigenfunctions $(-\Delta)^s\phi=\lambda\phi$ in $\Omega$, $\phi\equiv0$ in $\mathbb{R}^n\setminus\Omega$.
References:
[1] |
N. Abatangelo, Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian, preprint arXiv (Oct. 2013). |
[2] |
Y. Bozhkov and P. Olver, Pohozhaev and Morawetz identities in elastostatics and elastodynamics, SIGMA, 7 (2011), 055, 9pp.
doi: 10.3842/SIGMA.2011.055. |
[3] |
C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results, J. Funct. Anal., 263 (2012), 3741-3783.
doi: 10.1016/j.jfa.2012.09.006. |
[4] |
S.-Y. A. Chang and P. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.
doi: 10.4310/MRL.1997.v4.n1.a9. |
[5] |
K. S. Chou and X.-P. Zhu, Some constancy results for nematic liquid crystals and harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 99-115. |
[6] |
A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[7] |
A. Dalibard and D. Gérard-Varet, On shape optimization problems involving the fractional Laplacian, ESAIM Control Optim. Calc. Var., 19 (2013), 976-1013.
doi: 10.1051/cocv/2012041. |
[8] |
J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (2010), 1311-1333.
doi: 10.1007/s00023-009-0016-9. |
[9] |
B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555.
doi: 10.2478/s13540-012-0038-8. |
[10] |
S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-differential Equations of Parabolic Type, Birkhauser, Basel, 2004.
doi: 10.1007/978-3-0348-7844-9. |
[11] |
R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.
doi: 10.1090/S0002-9947-1961-0137148-5. |
[12] |
C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.
doi: 10.1007/s00222-002-0268-1. |
[13] |
G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528.
doi: 10.1016/j.aim.2014.09.018. |
[14] |
G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634.
doi: 10.1016/j.jmaa.2014.07.081. |
[15] |
N. Katz and N. Pavlovic, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[16] |
J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), 14-47.
doi: 10.2307/1971012. |
[17] |
J. L. Lions, Exact controllability, stabilization, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[18] |
R. L. Magin, O. Abdullah, D. Baleanu and X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Magnetic Resonance, 190 (2008), 255-270.
doi: 10.1016/j.jmr.2007.11.007. |
[19] |
T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local $Q$-curvature equation in dimension three, Calc. Var. Partial Differential Equations, (2014), 1-2.
doi: 10.1007/s00526-014-0718-9. |
[20] |
C. Miao, J. Yang and J. Zheng, An improved maximal inequality for 2D fractional order Schrödinger operators, preprint arXiv (Aug. 2013). |
[21] |
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[22] |
J. H. Ortega and E. Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation, SIAM J. Control Optim., 39 (2000), 1585-1614.
doi: 10.1137/S0363012900358483. |
[23] |
S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. |
[24] |
D. Pollack, Compactness results for complete metrics of constant positive scalar curvature on subdomains of $S^n$, Indiana Univ. Math. J., 42 (1993), 1441-1456.
doi: 10.1512/iumj.1993.42.42066. |
[25] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[26] |
F. Rellich, Darstellung der Eigenverte von $-\Delta u+\lambda u = 0$ durch ein Randintegral, Math. Z., 46 (1940), 635-636.
doi: 10.1007/BF01181459. |
[27] |
X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508.
doi: 10.1016/j.crma.2012.05.011. |
[28] |
X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[29] |
X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628.
doi: 10.1007/s00205-014-0740-2. |
[30] |
X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.
doi: 10.1080/03605302.2014.918144. |
[31] |
S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2002. |
[32] |
R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math., 41 (1988), 317-392.
doi: 10.1002/cpa.3160410305. |
[33] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[34] |
P. Sjölin, Regularity of solutions to the Schödinger equation, Duke Math. J., 55 (1987), 699-715.
doi: 10.1215/S0012-7094-87-05535-9. |
[35] |
W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series, Vol. 73, Amer. Math. Soc., Providence, R.I., 1989. |
[36] |
T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366.
doi: 10.2140/apde.2009.2.361. |
[37] |
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078.
doi: 10.2307/2374041. |
[38] |
R. van der Vorst, Variational identities and applications to differential systems, Arch. Rat. Mech. Anal., 116 (1992), 375-398.
doi: 10.1007/BF00375674. |
[39] |
R. Yang, On higher order extensions for the fractional Laplacian, preprint arXiv (Feb. 2013). |
[40] |
T. Zhu and J. M. Harris, Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians, Geophysics, 79 (2014), 1-12.
doi: 10.1190/geo2013-0245.1. |
show all references
References:
[1] |
N. Abatangelo, Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian, preprint arXiv (Oct. 2013). |
[2] |
Y. Bozhkov and P. Olver, Pohozhaev and Morawetz identities in elastostatics and elastodynamics, SIGMA, 7 (2011), 055, 9pp.
doi: 10.3842/SIGMA.2011.055. |
[3] |
C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results, J. Funct. Anal., 263 (2012), 3741-3783.
doi: 10.1016/j.jfa.2012.09.006. |
[4] |
S.-Y. A. Chang and P. Yang, On uniqueness of solutions of $n$-th order differential equations in conformal geometry, Math. Res. Lett., 4 (1997), 91-102.
doi: 10.4310/MRL.1997.v4.n1.a9. |
[5] |
K. S. Chou and X.-P. Zhu, Some constancy results for nematic liquid crystals and harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 12 (1995), 99-115. |
[6] |
A. Cotsiolis and N. K. Tavoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl., 295 (2004), 225-236.
doi: 10.1016/j.jmaa.2004.03.034. |
[7] |
A. Dalibard and D. Gérard-Varet, On shape optimization problems involving the fractional Laplacian, ESAIM Control Optim. Calc. Var., 19 (2013), 976-1013.
doi: 10.1051/cocv/2012041. |
[8] |
J. Dolbeault and R. Stanczy, Non-existence and uniqueness results for supercritical semilinear elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 10 (2010), 1311-1333.
doi: 10.1007/s00023-009-0016-9. |
[9] |
B. Dyda, Fractional calculus for power functions and eigenvalues of the fractional Laplacian, Fract. Calc. Appl. Anal., 15 (2012), 536-555.
doi: 10.2478/s13540-012-0038-8. |
[10] |
S. D. Eidelman, S. D. Ivasyshen and A. N. Kochubei, Analytic Methods in the Theory of Differential and Pseudo-differential Equations of Parabolic Type, Birkhauser, Basel, 2004.
doi: 10.1007/978-3-0348-7844-9. |
[11] |
R. K. Getoor, First passage times for symmetric stable processes in space, Trans. Amer. Math. Soc., 101 (1961), 75-90.
doi: 10.1090/S0002-9947-1961-0137148-5. |
[12] |
C. R. Graham and M. Zworski, Scattering matrix in conformal geometry, Invent. Math., 152 (2003), 89-118.
doi: 10.1007/s00222-002-0268-1. |
[13] |
G. Grubb, Fractional Laplacians on domains, a development of Hörmander's theory of $\mu$-transmission pseudodifferential operators, Advances in Mathematics, 268 (2015), 478-528.
doi: 10.1016/j.aim.2014.09.018. |
[14] |
G. Grubb, Spectral results for mixed problems and fractional elliptic operators, J. Math. Anal. Appl., 421 (2015), 1616-1634.
doi: 10.1016/j.jmaa.2014.07.081. |
[15] |
N. Katz and N. Pavlovic, A cheap Caffarelli-Kohn-Nirenberg inequality for the Navier-Stokes equation with hyper-dissipation, Geom. Funct. Anal., 12 (2002), 355-379.
doi: 10.1007/s00039-002-8250-z. |
[16] |
J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), 14-47.
doi: 10.2307/1971012. |
[17] |
J. L. Lions, Exact controllability, stabilization, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.
doi: 10.1137/1030001. |
[18] |
R. L. Magin, O. Abdullah, D. Baleanu and X. J. Zhou, Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation, J. Magnetic Resonance, 190 (2008), 255-270.
doi: 10.1016/j.jmr.2007.11.007. |
[19] |
T. Jin, A. Maalaoui, L. Martinazzi and J. Xiong, Existence and asymptotics for solutions of a non-local $Q$-curvature equation in dimension three, Calc. Var. Partial Differential Equations, (2014), 1-2.
doi: 10.1007/s00526-014-0718-9. |
[20] |
C. Miao, J. Yang and J. Zheng, An improved maximal inequality for 2D fractional order Schrödinger operators, preprint arXiv (Aug. 2013). |
[21] |
E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151.
doi: 10.1080/03605309308820923. |
[22] |
J. H. Ortega and E. Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation, SIAM J. Control Optim., 39 (2000), 1585-1614.
doi: 10.1137/S0363012900358483. |
[23] |
S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39. |
[24] |
D. Pollack, Compactness results for complete metrics of constant positive scalar curvature on subdomains of $S^n$, Indiana Univ. Math. J., 42 (1993), 1441-1456.
doi: 10.1512/iumj.1993.42.42066. |
[25] |
P. Pucci and J. Serrin, A general variational identity, Indiana Univ. Math. J., 35 (1986), 681-703.
doi: 10.1512/iumj.1986.35.35036. |
[26] |
F. Rellich, Darstellung der Eigenverte von $-\Delta u+\lambda u = 0$ durch ein Randintegral, Math. Z., 46 (1940), 635-636.
doi: 10.1007/BF01181459. |
[27] |
X. Ros-Oton and J. Serra, Fractional Laplacian: Pohozaev identity and nonexistence results, C. R. Math. Acad. Sci. Paris, 350 (2012), 505-508.
doi: 10.1016/j.crma.2012.05.011. |
[28] |
X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302.
doi: 10.1016/j.matpur.2013.06.003. |
[29] |
X. Ros-Oton and J. Serra, The Pohozaev identity for the fractional Laplacian, Arch. Rat. Mech. Anal., 213 (2014), 587-628.
doi: 10.1007/s00205-014-0740-2. |
[30] |
X. Ros-Oton and J. Serra, Nonexistence results for nonlocal equations with critical and supercritical nonlinearities, Comm. Partial Differential Equations, 40 (2015), 115-133.
doi: 10.1080/03605302.2014.918144. |
[31] |
S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2002. |
[32] |
R. Schoen, The existence of weak solutions with prescribed singular behavior for a conformally invariant scalar equation, Comm. Pure Appl. Math., 41 (1988), 317-392.
doi: 10.1002/cpa.3160410305. |
[33] |
L. Silvestre, Regularity of the obstacle problem for a fractional power of the laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[34] |
P. Sjölin, Regularity of solutions to the Schödinger equation, Duke Math. J., 55 (1987), 699-715.
doi: 10.1215/S0012-7094-87-05535-9. |
[35] |
W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series, Vol. 73, Amer. Math. Soc., Providence, R.I., 1989. |
[36] |
T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation, Anal. PDE, 2 (2009), 361-366.
doi: 10.2140/apde.2009.2.361. |
[37] |
K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078.
doi: 10.2307/2374041. |
[38] |
R. van der Vorst, Variational identities and applications to differential systems, Arch. Rat. Mech. Anal., 116 (1992), 375-398.
doi: 10.1007/BF00375674. |
[39] |
R. Yang, On higher order extensions for the fractional Laplacian, preprint arXiv (Feb. 2013). |
[40] |
T. Zhu and J. M. Harris, Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians, Geophysics, 79 (2014), 1-12.
doi: 10.1190/geo2013-0245.1. |
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