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May  2015, 35(5): 2131-2150. doi: 10.3934/dcds.2015.35.2131

Local integration by parts and Pohozaev identities for higher order fractional Laplacians

Xavier Ros-Oton 1, and  Joaquim Serra 2,

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

Received  June 2014 Revised  September 2014 Published  December 2014

We establish an integration by parts formula in bounded domains for the higher order fractional Laplacian $(-\Delta)^s$ with $s>1$. We also obtain the Pohozaev identity for this operator. Both identities involve local boundary terms, and they extend the identities obtained by the authors in the case $s\in(0,1)$.
    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 $\mathbb{R}^n\setminus\Omega$.
Keywords: Pohozaev identity, integration by parts, bounded domains, Dirichlet problem., higher order, Fractional Laplacian.
Mathematics Subject Classification: Primary: 47G30, 35S1.
Citation: Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131
References:
[1]

N. Abatangelo, Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian, preprint arXiv (Oct. 2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), 14-47. doi: 10.2307/1971012.  Google Scholar

[17]

J. L. Lions, Exact controllability, stabilization, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. Miao, J. Yang and J. Zheng, An improved maximal inequality for 2D fractional order Schrödinger operators, preprint arXiv (Aug. 2013). Google Scholar

[21]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[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.  Google Scholar

[23]

S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series, Vol. 73, Amer. Math. Soc., Providence, R.I., 1989.  Google Scholar

[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.  Google Scholar

[37]

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078. doi: 10.2307/2374041.  Google Scholar

[38]

R. van der Vorst, Variational identities and applications to differential systems, Arch. Rat. Mech. Anal., 116 (1992), 375-398. doi: 10.1007/BF00375674.  Google Scholar

[39]

R. Yang, On higher order extensions for the fractional Laplacian, preprint arXiv (Feb. 2013). Google Scholar

[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.  Google Scholar

show all references

References:
[1]

N. Abatangelo, Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian, preprint arXiv (Oct. 2013). Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

J. Kazdan and F. Warner, Curvature functions for compact 2-manifolds, Ann. of Math., 99 (1974), 14-47. doi: 10.2307/1971012.  Google Scholar

[17]

J. L. Lions, Exact controllability, stabilization, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68. doi: 10.1137/1030001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

C. Miao, J. Yang and J. Zheng, An improved maximal inequality for 2D fractional order Schrödinger operators, preprint arXiv (Aug. 2013). Google Scholar

[21]

E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations, 18 (1993), 125-151. doi: 10.1080/03605309308820923.  Google Scholar

[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.  Google Scholar

[23]

S. I. Pohozaev, On the eigenfunctions of the equation $\Delta u + \lambda f(u) = 0$, Dokl. Akad. Nauk SSSR, 165 (1965), 36-39.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

S. G. Samko, Hypersingular Integrals and Their Applications, Taylor and Francis, London, 2002.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

W. A. Strauss, Nonlinear Wave Equations, CBMS Regional Conference Series, Vol. 73, Amer. Math. Soc., Providence, R.I., 1989.  Google Scholar

[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.  Google Scholar

[37]

K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math., 98 (1976), 1059-1078. doi: 10.2307/2374041.  Google Scholar

[38]

R. van der Vorst, Variational identities and applications to differential systems, Arch. Rat. Mech. Anal., 116 (1992), 375-398. doi: 10.1007/BF00375674.  Google Scholar

[39]

R. Yang, On higher order extensions for the fractional Laplacian, preprint arXiv (Feb. 2013). Google Scholar

[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.  Google Scholar

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