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Bi-Laplacians on graphs and networks

  • Federica Gregorio
  • Delio MugnoloEmail author
Article

Abstract

We study the differential operator \(A=\frac{d^4}{dx^4}\) acting on a connected network \(\mathcal {G}\) along with \(\mathcal L^2\), the square of the discrete Laplacian acting on a connected discrete graph \(\mathsf {G}\). For both operators, we discuss well-posedness of the associated linear parabolic problems
$$\begin{aligned} \frac{\partial u}{\partial t}=-Au,\qquad \frac{df}{dt}=-{\mathcal {L}}^2 f, \end{aligned}$$
on \(L^p(\mathcal {G})\) or \(\ell ^p(\mathsf {V})\), respectively, for \(1\le p\le \infty \). In view of the well-known lack of parabolic maximum principle for all elliptic differential operators of order 2N for \(N>1\), our most surprising finding is that after some transient time, the parabolic equations driven by \(-A\) may display Markovian features, depending on the imposed transmission conditions in the vertices. Analogous results seem to be unknown in the case of general domains and even bounded intervals. Our analysis is based on a detailed study of bi-harmonic functions complemented by simple combinatorial arguments. We elaborate on analogous issues for the discrete bi-Laplacian; a characterization of complete graphs in terms of the Markovian property of the semigroup generated by \(-{\mathcal {L}}^2\) is also presented.

Keywords

Quantum graphs Differential and difference operators of higher order Positive semigroups of bounded linear operators Boundary conditions 

Mathematics Subject Classification

35K35 47D06 46G10 

Notes

References

  1. 1.
    W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander. Vector-Valued Laplace Transforms and Cauchy Problems, volume 96 of Monographs in Mathematics. Birkhäuser, Basel, 2001.Google Scholar
  2. 2.
    T. Ando and K. Nishio. Positive selfadjoint extensions of positive symmetric operators. Tokohu Math. J., 22:65–75, 1970.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    W. Arendt. Semigroups and evolution equations: Functional calculus, regularity and kernel estimates. In C.M. Dafermos and E. Feireisl, editors, Handbook of Differential Equations: Evolutionary Equations – Vol. 1. North Holland, Amsterdam, 2004.Google Scholar
  4. 4.
    W. Arendt. Heat Kernels – Manuscript of the \(9^{\rm th}\) Internet Seminar, 2006. (freely available at http://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/arendt/downloads/internetseminar.pdf).
  5. 5.
    W. Arendt and T. ter Elst. The Dirichlet-to-Neumann operator on \( C(\partial \Omega )\). arXiv:1707.05556, 2017.
  6. 6.
    A. Beurling and J. Deny. Dirichlet spaces. Proc. Natl. Acad. Sci. USA, 45:208–215, 1959.CrossRefzbMATHGoogle Scholar
  7. 7.
    J. von Below. A characteristic equation associated with an eigenvalue problem on \(c^2\)-networks. Lin. Algebra Appl., 71:309–325, 1985.CrossRefzbMATHGoogle Scholar
  8. 8.
    G. Berkolaiko and P. Kuchment. Introduction to Quantum Graphs, volume 186 of Math. Surveys and Monographs. Amer. Math. Soc., Providence, RI, 2013.Google Scholar
  9. 9.
    G. Berkolaiko, J.B. Kennedy, P. Kurasov, and D. Mugnolo. Surgery principles for the spectral analysis of quantum graphs. Trans. Amer. Math. Soc., (to appear).Google Scholar
  10. 10.
    A.V. Borovskikh and K.P. Lazarev. Fourth-order differential equations on geometric graphs. J. Math. Sci., 119:719–738, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    S. Bonaccorsi and S. Mazzucchi. High order heat-type equations and random walks on the complex plane. Stochastic Process. Appl., 125:797–818, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. Chen, M.C. Delfour, A.M. Krall, and G. Payre. Modeling, stabilization and control of serially connected beams. SIAM J. Control Opt., 25:526–546, 1987.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Courant, K. Friedrichs, and H. Lewy. Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann., 100:32–74, 1928.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Cardanobile and D. Mugnolo. Parabolic systems with coupled boundary conditions. J. Differ. Equ., 247:1229–1248, 2009.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    E.B. Davies. Long time asymptotics of fourth order parabolic equations. Journal d’Analyse Mathématique, 67:323–345, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E.B. Davies. Uniformly elliptic operators with measurable coefficients. J. Funct. Anal., 132:141–169, 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    E.B. Davies. Linear Operators And Their Spectra. Cambridge Univ. Press, Cambridge, 2007.CrossRefzbMATHGoogle Scholar
  18. 18.
    D. Daners, J. Glück, and J.B. Kennedy. Eventually and asymptotically positive semigroups on Banach lattices. J. Differ. Equ., 261:2607–2649, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    D. Daners, J. Glück, and J.B. Kennedy. Eventually positive semigroups of linear operators. J. Math. Anal. Appl., 433:1561–1593, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    R. Diestel. Graph Theory, volume 173 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2005.Google Scholar
  21. 21.
    R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2. Springer-Verlag, Berlin, 1988.CrossRefzbMATHGoogle Scholar
  22. 22.
    B. Dekoninck and S. Nicaise. Control of networks of Euler-Bernoulli beams. ESAIM: Control, Optimisation and Calculus of Variations, 4:57–81, 1999.Google Scholar
  23. 23.
    B. Dekoninck and S. Nicaise. The eigenvalue problem for networks of beams. Lin. Algebra Appl., 314:165–189, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    R. Dáger and E. Zuazua. Wave propagation, observation and control in 1-d flexible multi-structures, volume 50 of Mathém. & Appl. Springer-Verlag, Berlin, 2006.Google Scholar
  25. 25.
    K.-J. Engel. On singular perturbations of second order Cauchy problems. Pacific J. Math., 152:79–91, 1992.MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    A. Ferrero, F. Gazzola, and H.-C. Grunau. Decay and eventual local positivity for biharmonic parabolic equations. Disc. Cont. Dyn. Syst., 21:1129–1157, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    fedja (https://mathoverflow.net/users/1131/fedja). An elementary inequality for graph Laplacians. MathOverflow. https://mathoverflow.net/q/287765(version: 2017-12-05).
  28. 28.
    M. Fiedler. Algebraic connectivity of graphs. Czech. Math. J., 23:298–305, 1973.MathSciNetzbMATHGoogle Scholar
  29. 29.
    M. Fukushima, Y. Oshima, and M. Takeda. Dirichlet forms and symmetric Markov processes, volume 19 of Studies in Math. de Gruyter, Berlin, 2010.Google Scholar
  30. 30.
    T. Funaki. Probabilistic construction of the solution of some higher order parabolic differential equation. Proc. Japan Acad., Ser. A, 55:176–179, 1979.Google Scholar
  31. 31.
    F. Gazzola and H.-C. Grunau. Eventual local positivity for a biharmonic heat equation in \(\mathbb{R}^n\). Disc. Cont. Dyn. Syst. S, (83-87):265–266, 2008.zbMATHGoogle Scholar
  32. 32.
    R.J. Griego and R. Hersh. Random evolutions, Markov chains, and systems of partial differential equations. Proc. Natl. Acad. Sci. USA, 62:305–308, 1969.MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    S. Gnutzmann and F. Haake. Positivity violation and initial slips in open systems. Z. Phys. B, 101:263–273, 1996.CrossRefGoogle Scholar
  34. 34.
    L. Giacomelli, H. Knüpfer, and F. Otto. Smooth zero-contact-angle solutions to a thin-film equation around the steady state. J. Differ. Equ., 245:1454–1506, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    J. Glück. Invariant Sets and Long Time Behaviour of Operator Semigroups. PhD thesis, Universität Ulm, 2016.Google Scholar
  36. 36.
    J. Gasch and L. Maligranda. On vector-valued inequalities of the marcinkiewicz-zygmund, herz and krivine type. Math. Nachr., 167:95–129, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    F. Gregorio and S. Mildner. Fourth-order Schrödinger type operator with singular potentials. Arch. Math, 107:285–294, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    V.A. Galaktionov, E.L. Mitidieri, and S.I. Pohozaev. Blow-up for Higher-Order Parabolic, Hyperbolic, Dispersion and Schrodinger Equations. CRC Press, Boca Raton, FL, 2014.zbMATHGoogle Scholar
  39. 39.
    B. Gramsch. Zum Einbettungssatz von Rellich bei Sobolevräumen. Math. Z., 106:81–87, 1968.MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    S.M. Han, H. Benaroya, and T. Wei. Dynamics of transversely vibrating beams using four engineering theories. J. Sound Vib., 225:935–988, 1999.CrossRefzbMATHGoogle Scholar
  41. 41.
    O. Holtz and M. Karow. Real and complex operator norms. arXiv:math/0512608v1, 2005.
  42. 42.
    P.G. Hufton, Y.T. Lin, and T. Galla. Model reduction methods for classical stochastic systems with fast-switching environments: reduced master equations, stochastic differential equations, and applications. arXiv:1803.02941, 2018.
  43. 43.
    K.J. Hochberg. A signed measure on path space related to Wiener measure. Ann. Probab., 6:433–458, 1978.MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    E. Hölder. Entwicklungssätze aus der Theorie der zweiten Variation – Allgemeine Randbedingungen. Acta Math., 70:193–242, 1939.MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    T. Kato. On the semi-groups generated by Kolmogoroff’s differential equations. J. Math. Soc. Jap., 6:1–15, 1954.MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    J.B. Kennedy, P. Kurasov, G. Malenová, and D. Mugnolo. On the spectral gap of a quantum graph. Ann. Henri Poincaré, 17:2439–2473, 2016.MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    J.-C. Kiik, P. Kurasov, and M. Usman. On vertex conditions for elastic systems. Phys. Lett. A, 379:1871–1876, 2015.MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    R.V. Kohn and F. Otto. Upper bounds on coarsening rates. Commun. Math. Phys., 229:375–395, 2002.MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    M.G. Krein. The theory of self-adjoint extensions of semi-bounded hermitian transformations and its applications. I. Mat. Sbornik, 20:431–495, 1947.MathSciNetzbMATHGoogle Scholar
  50. 50.
    V.J. Krylov. Some properties of the distribution corresponding to the equation \(\partial u/\partial t=(-1)^{q+1}\partial ^{2q}u/\partial x^{2q}\). 1:760–763, 1960.Google Scholar
  51. 51.
    T. Kottos and U. Smilansky. Quantum chaos on graphs. Phys. Rev. Lett., 79:4794–4797, 1997.CrossRefGoogle Scholar
  52. 52.
    V. Kostrykin and R. Schrader. Kirchhoff’s rule for quantum wires. J. Phys. A, 32:595–630, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    P. Kuchment. Quantum graphs I: Some basic structures. Waves Random Media, 14:107–128, 2004.MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    J.E. Lagnese, G. Leugering, and E.J.P.G. Schmidt. Modelling and controllability of networks of thin beams. In System Modelling and Optimization, pages 467–480. Springer, 1992.Google Scholar
  55. 55.
    J.E. Lagnese, G. Leugering, and E.J.P.G. Schmidt. Modelling of dynamic networks of thin thermoelastic beams. Math. Meth. Appl. Sci., 16:327–358, 1993.MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    J.E. Lagnese, G. Leugering, and E.J.P.G. Schmidt. Modeling, Analysis, and Control of Dynamic Elastic Multi-Link Structures. Systems and Control: Foundations and Applications. Birkhäuser, Basel, 1994.Google Scholar
  57. 57.
    R.S. Laugesen and M.C. Pugh. Linear stability of steady states for thin film and Cahn-Hilliard type equations. Arch. Ration. Mech. Anal., 154:3–51, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    G. Lumer. Connecting of local operators and evolution equations on networks. In F. Hirsch, editor, Potential Theory (Proc. Copenhagen 1979), pages 230–243, Berlin, 1980. Springer-Verlag.Google Scholar
  59. 59.
    D. Mugnolo and R. Nittka. Properties of representations of operators acting between spaces of vector-valued functions. Positivity, 15:135–154, 2011.MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    B. Mohar. The spectrum of an infinite graph. Lin. Algebra Appl., 48:245–256, 1982.MathSciNetCrossRefzbMATHGoogle Scholar
  61. 61.
    A.H. Mueller, A.I. Shoshi, and S.M.H. Wong. Extension of the JIMWLK equation in the low gluon density region. Nuclear Phys. B, 715:440–460, 2005.CrossRefGoogle Scholar
  62. 62.
    D. Mugnolo. Gaussian estimates for a heat equation on a network. Networks Het. Media, 2:55–79, 2007.MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    D. Mugnolo. Parabolic theory of the discrete \(p\)-Laplace operator. Nonlinear Anal., Theory Methods Appl., 87:33–60, 2013.Google Scholar
  64. 64.
    D. Mugnolo. Semigroup Methods for Evolution Equations on Networks. Underst. Compl. Syst. Springer-Verlag, Berlin, 2014.CrossRefzbMATHGoogle Scholar
  65. 65.
    D. Mugnolo. Some remarks on the Krein-von Neumann extension of different Laplacians. In J. Banasiak, A. Bobrowski, and M. Lachowicz, editors, Semigroups of Operators-Theory and Applications, Proc. Math. & Stat., New York, 2015. Springer-Verlag.Google Scholar
  66. 66.
    A. Manavi, H. Vogt, and J. Voigt. Domination of semigroups associated with sectorial forms. J. Oper. Theory, 54:9–25, 2005.MathSciNetzbMATHGoogle Scholar
  67. 67.
    R. Nagel, editor. One-Parameter Semigroups of Positive Operators, volume 1184 of Lect. Notes Math. Springer-Verlag, Berlin, 1986.Google Scholar
  68. 68.
    S. Nicaise. Problémes de Cauchy posés en norme uniforme sur les espaces ramifiés élémentaires. C.R. Acad. Sc. Paris Sér. A, 303:443–446, 1986.Google Scholar
  69. 69.
    S. Nicaise. Spectre des réseaux topologiques finis. Bull. Sci. Math., II. Sér., 111:401–413, 1987.Google Scholar
  70. 70.
    R. Nittka. Projections onto convex sets and \(L^p\)-quasi-contractivity of semigroups. Arch. Math., 98:341–353, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  71. 71.
    E.M. Ouhabaz. Analysis of Heat Equations on Domains, volume 30 of Lond. Math. Soc. Monograph Series. Princeton Univ. Press, Princeton, NJ, 2005.Google Scholar
  72. 72.
    F. Rellich. Halbbeschränkte Differentialoperatoren höherer Ordnung. In Proc. Int. Congress of Mathematicians (Amsterdam 1954), volume 3, pages 243–250, Amsterdam, 1954. North-Holland.Google Scholar
  73. 73.
    J.-P. Roth. Spectre du laplacien sur un graphe. C. R. Acad. Sci. Paris Sér. I Math., 296:793–795, 1983.Google Scholar
  74. 74.
    K. Schmüdgen. Unbounded Self-adjoint Operators on Hilbert Space, volume 265 of Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2012.Google Scholar
  75. 75.
    A. Suárez, R. Silbey, and I. Oppenheim. Memory effects in the relaxation of quantum open systems. J. Chem. Phys., 97:5101–5107, 1992.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Lehrgebiet Analysis, Fakultät Mathematik und InformatikFernUniversität in HagenHagenGermany
  2. 2.Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica ApplicataUniversità degli Studi di SalernoFiscianoItaly

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