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

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 $\mathbb{R}^n\setminus\Omega$.
Citation: Xavier Ros-Oton, Joaquim Serra. Local integration by parts and Pohozaev identities for higher order fractional Laplacians. Discrete & Continuous Dynamical Systems - A, 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)., (2013).   Google Scholar

[2]

Y. Bozhkov and P. Olver, Pohozhaev and Morawetz identities in elastostatics and elastodynamics,, SIGMA, 7 (2011).  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.  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.  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.   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.  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.  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.  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.  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, (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.  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.  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.  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.  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.  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.  doi: 10.2307/1971012.  Google Scholar

[17]

J. L. Lions, Exact controllability, stabilization, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  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.  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.  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)., (2013).   Google Scholar

[21]

E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equations, 18 (1993), 125.  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.  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.   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.  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.  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.  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.  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.  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.  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.  doi: 10.1080/03605302.2014.918144.  Google Scholar

[31]

S. G. Samko, Hypersingular Integrals and Their Applications,, Taylor and Francis, (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.  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.  doi: 10.1002/cpa.20153.  Google Scholar

[34]

P. Sjölin, Regularity of solutions to the Schödinger equation,, Duke Math. J., 55 (1987), 699.  doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar

[35]

W. A. Strauss, Nonlinear Wave Equations,, CBMS Regional Conference Series, 73 (1989).   Google Scholar

[36]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361.  doi: 10.2140/apde.2009.2.361.  Google Scholar

[37]

K. Uhlenbeck, Generic properties of eigenfunctions,, Amer. J. Math., 98 (1976), 1059.  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.  doi: 10.1007/BF00375674.  Google Scholar

[39]

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

[2]

Y. Bozhkov and P. Olver, Pohozhaev and Morawetz identities in elastostatics and elastodynamics,, SIGMA, 7 (2011).  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.  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.  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.   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.  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.  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.  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.  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, (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.  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.  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.  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.  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.  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.  doi: 10.2307/1971012.  Google Scholar

[17]

J. L. Lions, Exact controllability, stabilization, and perturbations for distributed systems,, SIAM Rev., 30 (1988), 1.  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.  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.  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)., (2013).   Google Scholar

[21]

E. Mitidieri, A Rellich type identity and applications,, Comm. Partial Differential Equations, 18 (1993), 125.  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.  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.   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.  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.  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.  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.  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.  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.  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.  doi: 10.1080/03605302.2014.918144.  Google Scholar

[31]

S. G. Samko, Hypersingular Integrals and Their Applications,, Taylor and Francis, (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.  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.  doi: 10.1002/cpa.20153.  Google Scholar

[34]

P. Sjölin, Regularity of solutions to the Schödinger equation,, Duke Math. J., 55 (1987), 699.  doi: 10.1215/S0012-7094-87-05535-9.  Google Scholar

[35]

W. A. Strauss, Nonlinear Wave Equations,, CBMS Regional Conference Series, 73 (1989).   Google Scholar

[36]

T. Tao, Global regularity for a logarithmically supercritical hyperdissipative Navier-Stokes equation,, Anal. PDE, 2 (2009), 361.  doi: 10.2140/apde.2009.2.361.  Google Scholar

[37]

K. Uhlenbeck, Generic properties of eigenfunctions,, Amer. J. Math., 98 (1976), 1059.  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.  doi: 10.1007/BF00375674.  Google Scholar

[39]

R. Yang, On higher order extensions for the fractional Laplacian,, preprint arXiv (Feb. 2013)., (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.  doi: 10.1190/geo2013-0245.1.  Google Scholar

[1]

Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $ p $-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445

[2]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469

[3]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340

[4]

Xuefeng Zhang, Yingbo Zhang. Fault-tolerant control against actuator failures for uncertain singular fractional order systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 1-12. doi: 10.3934/naco.2020011

[5]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[6]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[7]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240

[8]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[9]

Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251

[10]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[11]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[12]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[14]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[15]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[16]

Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453

[17]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[18]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075

[19]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[20]

Yahia Zare Mehrjerdi. A new methodology for solving bi-criterion fractional stochastic programming. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020054

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (76)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]