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

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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

Abstract
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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 and Continuous Dynamical Systems, 2015, 35 (5) : 2131-2150. doi: 10.3934/dcds.2015.35.2131

References:

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[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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