Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples

doi:10.1017/CBO9781139135061.007

6 - Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples

Published online by Cambridge University Press: 05 November 2012

J. Behrndt and

M. Langer

Edited by

Seppo Hassi ,

Hendrik S. V. de Snoo and

Franciszek Hugon Szafraniec

Show author details

J. Behrndt: Affiliation:
Technische Universität Graz
M. Langer: Affiliation:
University of Strathclyde
Seppo Hassi: Affiliation:
University of Vaasa, Finland
Hendrik S. V. de Snoo: Affiliation:
Rijksuniversiteit Groningen, The Netherlands
Franciszek Hugon Szafraniec: Affiliation:
Jagiellonian University, Krakow

Book contents

Get access

Summary

Abstract The notion of quasi boundary triples and their Weyl functions is reviewed and applied to self-adjointness and spectral problems for a class of elliptic, formally symmetric, second order partial differential expressions with variable coefficients on bounded domains.

Introduction

Boundary triples and associated Weyl functions are a powerful and ef- fficient tool to parameterize the self-adjoint extensions of a symmetric operator and to describe their spectral properties. There are numerous papers applying boundary triple techniques to spectral problems for various types of ordinary differential operators in Hilbert spaces; see, e.g. [Behrndt and Langer, 2010; Behrndt, Malamud and Neidhardt, 2008; Behrndt and Trunk, 2007; Brasche, Malamud and Neidhardt, 2002; Brüning, Geyler and Pankrashkin, 2008; Derkach, Hassi and de Snoo, 2003; Gorbachuk and Gorbachuk, 1991; Derkach and Malamud, 1995; Karabash, Kostenko and Malamud, 2009; Kostenko and Malamud, 2010; Posilicano, 2008] and the references therein.

The abstract notion of boundary triples and Weyl functions is strongly inspired by Sturm-Liouville operators on a half-line and their Titchmarsh -Weyl coefficients. To make this more precise, let us consider the ordinary differential expression l = −D2 + q on the positive half-line ℝ+ = (0, ∞), where D denotes the derivative, and suppose that q is a real-valued L∞-function. The maximal operator associated with l in L2(ℝ+) is defined on the Sobolev space H2(ℝ+) and turns out to be the adjoint of the minimal operator S f = l(f), dom S =, where is the subspace of H2(ℝ+) consisting of functions f that satisfy the boundary conditions f(0) = f′(0) = 0.

Information

Type: Chapter
Information: Operator Methods for Boundary Value Problems , pp. 121 - 160

DOI: https://doi.org/10.1017/CBO9781139135061.007 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Book purchase

Temporarily unavailable

References

Arov, D. Z., Kurula, M., and Staffans, O. J. 2011. Canonical State/Signal Shift Realizations of Passive Continuous Time Behaviors. Complex Anal. Oper. Theory, 5, 331–402.Google Scholar

Arov, D. Z., and Nudelman, M. A. 1996. Passive linear stationary dynamical scattering systems with continuous time. Integral Equations Operator Theory, 24, 1–45.Google Scholar

Arov, D. Z., and Staffans, O. J. 2007a. State/signal linear time-invariant systems theory. Part III: Transmission and impedance representations of discrete time systems. Pages 101–140 of: Operator Theory, Structured Matrices, and Dilations, Tiberiu Constantinescu Memorial Volume. Bucharest Romania: Theta Foundation. available from American Mathematical Society.

Arov, D. Z., and Staffans, O. J. 2007b. State/signal linear time-invariant systems theory. Part IV: Affine representations of discrete time systems. Complex Anal. Oper. Theory, 1, 457–521.Google Scholar

Arov, D. Z., and Staffans, O. J. 2012. Symmetries in special classes of passive state/signal systems. J. Funct. Anal., to appear.Google Scholar

Ball, J. A., and Staffans, O. J. 2006. Conservative state-space realizations of dissipative system behaviors. Integral Equations Operator Theory, 54, 151–213.Google Scholar

Behrndt, J., Hassi, S., and de Snoo, H. S. V. 2009. Boundary relations, unitary colligations, and functional models. Complex Anal. Oper. Theory, 3, 57–98.Google Scholar

Curtain, R. F., and Weiss, G. 1989. Well posedness of triples of operators (in the sense of linear systems theory). Pages 41–59 of: Control and Optimization of Distributed Parameter Systems. International Series of Numerical Mathematics, vol. 91. Basel Boston Berlin: Birkhäuser-Verlag.

Derkach, V. 2009. Abstract interpolation problem in Nevanlinna classes. Pages 197–236 of: Modern analysis and applications. The Mark Krein Centenary Conference. Vol. 1: Operator theory and related topics. Oper. Theory Adv. Appl., vol. 190. Basel: Birkhäuser Verlag.

Derkach, V. A., Hassi, S., Malamud, M. M., and de Snoo, H. S. V. 2006. Boundary relations and their Weyl families. Trans. Amer. Math. Soc., 358, 5351–5400.Google Scholar

Derkach, V. A., Hassi, S., Malamud, M. M., and de Snoo, H. S. V. 2009. Boundary relations and generalized resolvents of symmetric operators. Russ. J. Math. Phys., 16, 17–60.Google Scholar

Kurula, M. 2010. On passive and conservative state/signal systems in continuous time. Integral Equations Operator Theory, 67, 377–424, 449.Google Scholar

Kurula, M. 2010b. Towards input/output-free modelling of linear infinite-dimensional systems in continuous time. Ph.D. thesis, ISBN 978-952-12-2410-2, 230 pages, electronic summary http://urn.fi/URN:ISBN:978-952-12-2418-8.

Kurula, M., Zwart, H., van der Schaft, A., and Behrndt, J. 2010. Dirac structures and their composition on Hilbert spaces. J. Math. Anal. Appl., 372, 402–422.Google Scholar

Kurula, M., and Staffans, O. J. 2009. Well-posed state/signal systems in continuous time. Complex Anal. Oper. Theory, 4, 319–390.Google Scholar

Kurula, M., and Staffans, O. J. 2011. Connections between smooth and generalized trajectories of a state/signal system. Complex Anal. Oper. Theory, 5, 403–422.Google Scholar

Malinen, J., and Staffans, O. J. 2006. Conservative boundary control systems. J. Differential Equations, 231, 290–312.Google Scholar

Pazy, A. 1983. Semi-Groups of Linear Operators and Applications to Partial Differential Equations. Berlin: Springer-Verlag.

Salamon, D. 1987. Infinite dimensional linear systems with unbounded control and observation: a functional analytic approach. Trans. Amer. Math. Soc., 300, 383–431.Google Scholar

Šmuljan, Yu. L. 1986. Invariant subspaces of semigroups and the Lax·–Phillips scheme. Deposited in VINITI, No. 8009-B86, Odessa, 49 pages.

Staffans, O. J. 2002a. Passive and conservative continuous-time impedance and scattering systems. Part I: Well-posed systems. Math. Control Signals Systems, 15, 291–315.Google Scholar

Staffans, O. J. 2002b. Passive and conservative in?nite-dimensional impedance and scattering systems (from a personal point of view). Pages 375–414 of: Mathematical Systems Theory in Biology, Communication, Computation, and Finance. IMA Volumes in Mathematics and its Applications, vol. 134. New York: Springer-Verlag.

Staffans, O. J. 2005. Well-Posed Linear Systems. Cambridge and New York: Cambridge University Press.

Tsekanovskiĭ, E. R., and Šmuljan, Yu. L. 1977. The theory of biextensions of operators in rigged Hilbert spaces. Unbounded operator colligations and characteristic functions. Uspehi Mathem. Nauk SSSR, 32, 69–124.Google Scholar

Allakhverdiev, B. P. 1991. On the theory of dilatation and on the spectral analysis of dissipative Schrodinger operators in the case of the Weyl limit circle. Math. USSR-Izv., 36, 247–262.Google Scholar

Alpay, D., and Behrndt, J. 2009. Generalized Q-functions and Dirichlet-to-Neumann maps for elliptic differential operators. J. Funct. Anal., 257, 1666–1694.Google Scholar

Amrein, W. O., and Pearson, D. B. 2004. M-operators: a generalisation of Weyl-Titchmarsh theory. J. Comput. Appl. Math., 171, 1–26.Google Scholar

Arendt, W., and ter Elst, A. F. M. 2011. The Dirichlet-to-Neumann operator on rough domains. J. Differential Equations, 251, 2100–2124.Google Scholar

Arlinskiĭ, Yu. 2000. Abstract boundary conditions for maximal sectorial extensions of sectorial operators. Math. Nachr., 209, 5–36.Google Scholar

Ashbaugh, M. S., Gesztesy, F., Mitrea, M., and Teschl, G. 2010. Spectral theory for perturbed Krein Laplacians in nonsmooth domains. Adv. Math. 223, 1372–1467.Google Scholar

Astala, K., and Paivarinta, L. 2006. Calderon's inverse conductivity problem in the plane. Ann. of Math. (2), 163, 265–299.Google Scholar

Azizov, T. Ya., and Iokhvidov, I. S. 1989. Linear Operators in Spaces with an Indefinite Metric. Pure and Applied Mathematics (New York), John Wiley and Sons, Chichester.

Behrndt, J. 2010. Elliptic boundary value problems with A-dependent boundary conditions. J. Differential Equations, 249, 2663–2687.Google Scholar

Behrndt, J., Hassi, S., de Snoo, H.S.V., and Wietsma, H.L. 2011. Square-integrable solutions and Weyl functions for singular canonical systems. Math. Nachr., 284, 1334–1384.Google Scholar

Behrndt, J., and Langer, M. 2007. Boundary value problems for elliptic partial differential operators on bounded domains. J. Funct. Anal., 243, 536–565.Google Scholar

Behrndt, J., and Langer, M. 2010. On the adjoint of a symmetric operator. J. London Math. Soc. (2), 82, 563–580.Google Scholar

Behrndt, J., Langer, M., Lobanov, I., Lotoreichik, V., and Popov, I. Yu. 2010. A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains. J. Math. Anal. Appl., 371, 750–758.Google Scholar

Behrndt, J., Langer, M., and Lotoreichik, V. 2011. Spectral estimates for resolvent differences of self-adjoint elliptic operators. Submitted; preprint: arXiv:1012.4596v1 [math.SP]

Behrndt, J., Malamud, M. M., Neidhardt, H. 2008. Scattering matrices and Weyl functions. Proc. London Math. Soc. (33), 97, 568–598.Google Scholar

Behrndt, J., and Trunk, C. 2007. On the negative squares of indefinite Sturm-Liouville operators. J. Differential Equations, 238, 491–519.Google Scholar

Birman, M. Sh. 1962. Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions (Russian). Vestnik Le. Univ., 17, 22–55, (translation in Amer. Math. Soc. Transl., 225 (2008), 19–53).Google Scholar

Birman, M. Š., and Solomjak, M. Z. 1980. Asymptotic behavior of the spectrum of variational problems on solutions of elliptic equations in unbounded domains (Russian). Funktsional. Anal. i Prilozhen., 14, 27–35Google Scholar

(translation in Funct. Anal. Appl., 14 (1981), 267–274).

Brasche, J., Malamud, M. M., and Neidhardt, H. 2002. Weyl function and spectral properties of selfadjoint extensions. Integral Equations Operator Theory, 43, 264–289.Google Scholar

Brown, B. M., Grubb, G., and Wood, I. G. 2009. M-functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Math. Nachr., 282, 314–347.Google Scholar

Brown, M., Hinchcliffe, J., Marletta, M., Naboko, S., and Wood, I. 2009. The abstract Titchmarsh-Weyl M-function for adjoint operator pairs and its relation to the spectrum. Integral Equations Operator Theory, 63, 297–320.Google Scholar

Brown, B. M., and Marletta, M. 2004. Spectral inclusion and spectral exactness for PDEs on exterior domains. IMAJ.Numer.Anal., 24, 21–43.Google Scholar

Brown, M., Marletta, M., Naboko, S., and Wood, I. 2008. Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices. J. Lond. Math. Soc. (2), 77, 700–718.Google Scholar

Bruk, V. M. 1976. A certain class of boundary value problems with a spectral parameter in the boundary condition (Russian). Mat. Sb., 100 (142), 210–216.Google Scholar

Brüning, J., Geyler, V., and Pankrashkin, K. 2008. Spectra of self-adjoint extensions and applications to solvable Schrodinger operators. Rev. Math. Phys., 20, 1–70.Google Scholar

Derkach, V. A., Hassi, S., Malamud, M. M., and de Snoo, H. S. V. 2006. Boundary relations and their Weyl families. Trans. Amer. Math. Soc., 358, 5351–5400.Google Scholar

Derkach, V. A., Hassi, S., Malamud, M. M., and de Snoo, H. S. V. 2009. Boundary relations and generalized resolvents of symmetric operators. Russ. J. Math. Phys., 16, 17–60.Google Scholar

Derkach, V. A., Hassi, S., and de Snoo, H. S. V. 2003. Singular perturbations of self-adjoint operators. Math. Phys. Anal. Geom., 6, 349–384.Google Scholar

Derkach, V. A., and Malamud, M. M. 1991. Generalized resolvents and the boundary value problems for Hermitian operators with gaps. J. Funct. Anal., 95, 1–95.Google Scholar

Derkach, V. A., and Malamud, M. M. 1995. The extension theory of Hermitian operators and the moment problem. J. Math. Sciences, 73, 141–242.Google Scholar

Edmunds, D. E., and Evans, W. D. 1987. Spectral Theory and Differential Operators. The Clarendon Press, Oxford University Press, New York.

Everitt, W. N., and Markus, L. 2003. Elliptic partial differential operators and sym-plectic algebra. Mem. Amer. Math. Soc., 162, no. 770, 111 pp.Google Scholar

Everitt, W. N., Markus, L., and Plum, M. 2005. An unusual self-adjoint linear partial differential operator. Trans. Amer. Math. Soc., 357, 1303–1324.Google Scholar

Filonov, N. 2004. On an inequality for the eigenvalues of the Dirichlet and Neumann problems for the Laplace operator (Russian). Algebra i Analiz, 16, 172–176Google Scholar

(translation in St. Petersburg Math. J., 16 (2005), 413–416).

Friedlander, L. 1991. Some inequalities between Dirichlet and Neumann eigenvalues. Arch. Rational Mech. Anal., 116, 153–160.Google Scholar

Gesztesy, F., Makarov, K. A., and Tsekanovskii, E. 1998. An addendum to Krein's formula. J. Math. Anal. Appl., 222, 594–606.Google Scholar

Gesztesy, F., and Mitrea, M. 2008. Generalized Robin boundary conditions, Robin-to-Dirichlet maps, and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. Pages 105-173 of: Perspectives in Partial Differential Equations, Harmonic Analysis and Applications. Proc. Sympos. Pure Math., Vol. 79, Amer. Math. Soc., Providence, RI.

Gesztesy, F., and Mitrea, M. 2009. Robin-to-Robin maps and Krein-type resolvent formulas for Schrödinger operators on bounded Lipschitz domains. Pages 81-113 of: Modern Analysis and Applications. The Mark Krein Centenary Conference. Vol. 2: Differential Operators and Mechanics. Oper. Theory Adv. Appl., Vol. 191, Birkhäauser Verlag, Basel.

Gesztesy, F., and Mitrea, M. 2011. A description of all self-adjoint extensions of the Laplacian and Krein-type resolvent formulas on non-smooth domains. J. Anal. Math., 113, 53–172.Google Scholar

Gesztesy, F., and Tsekanovskii, E. 2000. On matrix-valued Herglotz functions. Math. Nachr., 218, 61–138.Google Scholar

Gorbachuk, V. I., and Gorbachuk, M. L. 1991. Boundary Value Problems for Operator Differential Equations. Mathematics and its Applications (Soviet Series), Vol. 48, Kluwer Academic Publishers, Dordrecht.

Grubb, G. 1968. A characterization of the non-local boundary value problems associated with an elliptic operator. Ann. Scuola Norm. Sup. Pisa (3), 22, 425–513.Google Scholar

Grubb, G. 1971. On coerciveness and semiboundedness of general boundary problems. Israel J. Math., 10, 32–95.Google Scholar

Grubb, G. 1974. Properties of normal boundary problems for elliptic even-order systems. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 1, 1–61.Google Scholar

Grubb, G. 1983. Spectral asymptotics for the “soft” selfadjoint extension of a symmetric elliptic differential operator. J. Operator Theory, 10, 9–20.Google Scholar

Grubb, G. 1984a. Singular Green operators and their spectral asymptotics. Duke Math. J., 51, 477–528.Google Scholar

Grubb, G. 1984b. Remarks on trace estimates for exterior boundary problems. Comm. Partial Differential Equations, 9, 231–270.Google Scholar

Grubb, G. 2006. Known and unknown results on elliptic boundary problems. Bull. Amer. Math. Soc., 43, 227–230.Google Scholar

Grubb, G. 2008. Krein resolvent formulas for elliptic boundary problems in nons-mooth domains. Rend. Semin. Mat. Univ. Politec. Torino, 66, 13–39.Google Scholar

Grubb, G. 2009. Distributions and Operators. Graduate Texts in Mathematics 252, Springer, New York.

Grubb, G. 2011. Spectral asymptotics for Robin problems with a discontinuous coefficient. J. Spectral Theory, 1, 155–177.Google Scholar

Kac, I. S., and Kreĭn, M. G. 1974. R-functions — analytic functions mapping the upper halfplane into itself. Amer. Math. Soc. Transl. (2), 103, 1–18.Google Scholar

Karabash, I. M., Kostenko, A. S., and Malamud, M. M. 2009. The similarity problem for J-nonnegative Sturm-Liouville operators. J. Differential Equations, 246, 964–997.Google Scholar

Kochubei, A. N. 1975. On extensions of symmetric operators and symmetric binary relations (Russian). Mat. Zametki, 17, 41–48.Google Scholar

Kostenko, A. S., and Malamud, M. M. 2010. 1-D Schröadinger operators with local point interactions on a discrete set. J. Differential Equations, 249, 253–304.Google Scholar

Kopachevskii, N. D., and Krein, S. G. 2004. Abstract Green formula for a triple of Hilbert spaces, abstract boundary-value and spectral problems (Russian). Ukr. Mat. Visn., 1, 69–97Google Scholar

(translation in Ukr. Math. Bull., 1 (2004), 77–105).

Kreĭn, M. G. 1946. Concerning the resolvents of an Hermitian operator with the deficiency-index (m, m). C. R. (Doklady) Acad. Sci. URSS (N.S.), 52, 651–654.Google Scholar

Langer, H., and Textorius, B. 1977. On generalized resolvents and Q-functions of symmetric linear relations (subspaces) in Hilbert space. Pacific J. Math., 72, 135–165.Google Scholar

Lions, J., and Magenes, E. 1972. Non-Homogeneous Boundary Value Problems and Applications I. Springer Verlag, New York – Heidelberg.

Malamud, M. M. 1992. On a formula for the generalized resolvents of a non-densely defined Hermitian operator (Russian). Ukrain. Mat. Zh., 44, 1658–1688Google Scholar

(translation in Ukrainian Math. J., 44 (1992), 1522–1547 (1993)).

Malamud, M. M. 2010. Spectral theory of elliptic operators in exterior domains. Russ. J. Math. Phys., 17, 96–125.Google Scholar

Malinen, J., and Staffans, O. J. 2007. Impedance passive and conservative boundary control systems. Complex Anal. Oper. Theory, 1, 279–300.Google Scholar

Marletta, M. 2004. Eigenvalue problems on exterior domains and Dirichlet to Neumann maps. J. Comput. Appl. Math., 171, 367–391.Google Scholar

Marletta, M. 2010. Neumann-Dirichlet maps and analysis of spectral pollution for non-self-adjoint elliptic PDEs with real essential spectrum. IMA J. Numer. Anal. 30, 917–939.Google Scholar

Mogilevskii, V. 2006. Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers. Methods Funct. Anal. Topology, 12, 258–280.Google Scholar

Mogilevskiĭ, V. 2009. Boundary triplets and Titchmarsh-Weyl functions of differential operators with arbitrary deficiency indices. Methods Funct. Anal. Topology, 15, 280–300.Google Scholar

Nachman, A. I. 1988. Reconstructions from boundary measurements. Ann. of Math. (2), 128, 531–576.Google Scholar

Nachman, A. I. 1996. Global uniqueness for a two-dimensional inverse boundary value problem. Ann. of Math. (2) 143, 71–96.Google Scholar

Nachman, A. I., Sylvester, J., and Uhlmann, G. 1988. An ri-dimensional Borg-Levinson theorem. Comm. Math. Phys., 115, 595–605.Google Scholar

Pankrashkin, K. 2006. Resolvents of self-adjoint extensions with mixed boundary conditions. Rep. Math. Phys., 58, 207–221.Google Scholar

Posilicano, A. 2004. Boundary triples and Weyl functions for singular perturbations of self-adjoint operators. Methods Funct. Anal. Topology, 10, 57–63.Google Scholar

Posilicano, A. 2008. Self-adjoint extensions of restrictions. Oper. Matrices, 2, 483–506.Google Scholar

Posilicano, A., and Raimondi, L. 2009. Krein's resolvent formula for self-adjoint extensions of symmetric second-order elliptic differential operators. J. Phys. A, 42, 015204, 11 pp.Google Scholar

Post, O. 2007. First-order operators and boundary triples. Russ. J. Math. Phys., 14, 482–492.Google Scholar

Ryzhov, V. 2007. A general boundary value problem and its Weyl function. Opuscula Math., 27, 305–331.Google Scholar

Ryzhov, V. 2009. Weyl-Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control. Complex Anal. Oper. Theory, 3, 289–322.Google Scholar

Saakjan, Š. N. 1965. Theory of resolvents of a symmetric operator with infinite defect numbers (Russian). Acad. Nauk Armjan. SSR Dokl., 41, 193–198.Google Scholar

Safarov, Y. 2008. On the comparison of the Dirichlet and Neumann counting functions. Amer. Math. Soc. Transl. Ser. 2, 225, pp. 191–204, Amer. Math. Soc., Providence, RI.

Seeley, R. 1969. The resolvent of an elliptic boundary problem. Amer. J. Math., 91, 889–920.Google Scholar

Šmuljan, Ju. L. 1976. Theory of linear relations, and spaces with indefinite metric (Russian). Funkcional. Anal. i Priložen., 10, 67–72.Google Scholar

Sylvester, J., and Uhlmann, G. 1987. A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. (2) 125, 153–169.Google Scholar

Višik, M. I. 1952. On general boundary problems for elliptic differential equations (Russian). Trudy Moskov. Mat. Obšč., 1, 187–246Google Scholar

(translation in Amer. Math. Soc. Transl., 24 (1963), 107–172).

Wloka, J. 1987. Partial Differential Equations. Cambridge University Press, Cambridge.

Accessibility standard: Unknown

Why this information is here

This section outlines the accessibility features of this content - including support for screen readers, full keyboard navigation and high-contrast display options. This may not be relevant for you.

Accessibility Information

Accessibility compliance for the PDF of this chapter is currently unknown and may be updated in the future.