Mathematics

# Dr Ian Wood

Lecturer in Mathematics

SMSAS - Mathematics Group

Room: RA104 (Rutherford Annexe)
Modules taught:
MA552: Analysis
MA577: Elements of Abstract Analysis
MA599/600/601: Mathematics Mini-Projects/Dissertation/Individual Projects

Office hours

Research Interests: Analysis of PDEs and spectral theory, in particular study of spectral properties of non-selfadjoint operators via boundary triples and M-functions (generalised Dirichlet-to-Neumann maps), regularity to solutions of PDEs in Lipschitz domains and waveguides in periodic structures.

Ian is Deputy Examinations Officer for undergraduate Mathematics programmes and serves on the School's Research Postgraduate Staff Student Consultative Committee.

Also view these in the Kent Academic Repository
Books

Wood, Ian (2005) Elliptic and Parabolic Problems in Non-Smooth Domains. Logos-Verlag, Berlin, 136 pp. ISBN 978-3-8325-1059-6.

Abstract

Regularity of solutions is an important part of the theory of partial differential equations. In this text, the regularity of solutions to elliptic and parabolic problems in Lipschitz domains is investigated. Maximal regularity estimates are useful when dealing with nonlinear parabolic problems. However, the known maximal regularity results for smooth domains no longer hold in Lp-spaces over Lipschitz domains for the whole range of exponents p. Here, maximal regularity estimates are shown for the Laplacian with suitable domain in Lp-spaces for a restricted range of p. Operators with L?-coefficients in convex domains and Ornstein-Uhlenbeck operators in exterior Lipschitz domains are also discussed.

Articles

Abels, Helmut and Grubb, Gerd and Wood, Ian (2014) Extension Theory and Krein-type Resolvent Formulas for Nonsmooth Boundary Value Problems. Journal of Functional Analysis, 266 (7). pp. 4037-4100. ISSN 0022-1236.

Abstract

For a strongly elliptic second-order operator $A$ on a bounded domain $\Omega\subset \mathbb{R}^n$ it has been known for many years how to interpret the general closed $L_2(\Omega)$-realizations of $A$ as representing boundary conditions (generally nonlocal), when the domain and coefficients are smooth. The purpose of the present paper is to extend this representation to nonsmooth domains and coefficients, including the case of H\"older $C^{\frac32+\varepsilon}$-smoothness, in such a way that pseudodifferential methods are still available for resolvent constructions and ellipticity considerations. We show how it can be done for domains with $B^\frac32_{2,p}$-smoothness and operators with $H^1_q$-coefficients, for suitable $p>2(n-1)$ and $q>n$. In particular, Kre\u\i{}n-type resolvent formulas are established in such nonsmooth cases. Some unbounded domains are allowed.

Brown, Brian Malcolm and Hoang, Vu and Plum, Michael et al. (2014) Spectrum created by line defects in periodic structures. Mathematische Nachrichten, 287 (17-18). pp. 1972-1985. ISSN 0025-584X.

Abstract

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a three-dimensional periodic medium; the defect is infinitely extended in one direction, but compactly supported in the remaining two. This perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. We will show that even small perturbations lead to additional spectrum in the spectral gaps of the unperturbed operator and investigate some properties of the spectrum that is created.

Brown, Brian Malcolm and Hoang, Vu and Plum, Michael et al. (2012) On the spectrum of waveguides in planar photonic bandgap structures. arxiv.org. pp. 1-19.

Abstract

We study a Helmholtz-type spectral problem related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a two-dimensional periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. The perturbation is expected to introduce guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem. In the first part of the paper, we prove that, somewhat unexpectedly, guided mode spectrum can be created by arbitrarily "small" perturbations. Secondly we show that, after performing a Floquet decomposition in the axial direction of the waveguide, for any fixed value of the quasi-momentum $k_x$ the perturbation generates at most finitely many new eigenvalues inside the gap.

Brown, Brian Malcolm and Grubb, Gerd and Wood, Ian (2009) M -functions for closed extensions of adjoint pairs of operators with applications to elliptic boundary problems. Mathematische Nachrichten, 282 (3). pp. 314-347. ISSN 0025-584X.

Abstract

In this paper, we combine results on extensions of operators with recent results on the relation between the M -function and the spectrum, to examine the spectral behaviour of boundary value problems. M -functions are defined for general closed extensions, and associated with realisations of elliptic operators. In particular, we consider both ODE and PDE examples where it is possible for the operator to possess spectral points that cannot be detected by the M -function

Brown, Brian Malcolm and Hinchcliffe, James and Marletta, Marco et al. (2009) The Abstract TITCHMARSH-WEYL M-Function for adjoint operator pairs and its relation to the Spectrum. Integral Equations and Operator Theory, 63 (3). pp. 297-320. ISSN 0378-620X.

Abstract

In the setting of adjoint pairs of operators we consider the question: to what extent does the Weyl M-function see the same singularities as the resolvent of a certain restriction AB of the maximal operator? We obtain results showing that it is possible to describe explicitly certain spaces S and S such that the resolvent bordered by projections onto these subspaces is analytic everywhere that the M-function is analytic. We present three examples – one involving a Hain-Lüst type operator, one involving a perturbed Friedrichs operator and one involving a simple ordinary differential operators on a half line – which together indicate that the abstract results are probably best possible.

Brown, Brian Malcolm and Eastham, M.S.P. and Wood, Ian (2009) Conditions for the spectrum associated with anasymptotically straight leaky wire to comprisethe interval (??, ?). Journal of Physics A: Mathematical and Theoretical, 42 (5). ISSN 1751-8113.

Abstract

We consider a quantum (or leaky) wire in the plane, and the wire supports a singular attraction which becomes large at distant points on the wire. An analogous regular potential arises from the motion of a hydrogen atom in an electric field. We prove that, as in the regular case, the spectrum is the whole of (??, ?).

Brown, Brian Malcolm and Eastham, M.S.P. and Wood, Ian (2009) Estimates for the lowest eigenvalue of a star graph. Journal of Mathematical Analysis and Applications, 354 (1). pp. 24-30. ISSN 0022-247X.

Abstract

We derive new estimates for the lowest eigenvalue of the Schrödinger operator associated with a star graph in R^2. We achieve this by a variational method and a procedure for identifying test functions which are sympathetic to the geometry of the star graph.

Brown, Brian Malcolm and Eastham, M.S.P. and Wood, Ian (2008) Conditions for the spectrum associated with an asymptotically straight leaky wire to contain an interval [? ?^2/4, ?). Archiv der Mathematik, 90 (6). pp. 554-558. ISSN 0003-889X.

Abstract

The method of singular sequences is used to provide a simple and, in some respects, a more general proof of a known spectral result for leaky wires. The method introduces a new concept of asymptotic straightness.

Brown, Brian Malcolm and Marletta, Marco and Naboko, Serguei et al. (2008) Boundary triplets and M-functions for non-selfadjoint operators, with applications to elliptic PDES and block operator matrices. Journal of the London Mathematical Society, 77 (3). pp. 700-718. ISSN 0024-6107.

Abstract

Starting with an adjoint pair of operators, under suitable abstract versions of standard PDE hypotheses, we consider the Weyl M-function of extensions of the operators. The extensions are determined by abstract boundary conditions and we establish results on the relationship between the M-function as an analytic function of a spectral parameter and the spectrum of the extension. We also give an example where the M-function does not contain the whole spectral information of the resolvent, and show that the results can be applied to elliptic PDEs where the M-function corresponds to the Dirichlet to Neumann map.

Wood, Ian (2007) Maximal L^p -regularity for the Laplacian on Lipschitz domains. Mathematische Zeitschrift, 255 (4). pp. 855-875. ISSN 0025-5874.

Abstract

We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains ?, both with the following two domains of definition:D1(?) = {u ? W1,p(?) : ?u ? Lp(?), Bu = 0}, orD2(?) = {u ? W2,p(?) : Bu = 0}, where B is the boundary operator.We prove that, under certain restrictions on the range of p, these operators generate positive analytic contraction semigroups on Lp(?) which implies maximal regularity for the corresponding Cauchy problems. In particular, if ? is bounded and convex and 1 < p ? 2, the Laplacian with domain D2(?) has the maximal regularity property, as in the case of smooth domains. In the last part,we construct an example that proves that, in general, the Dirichlet–Laplacian with domain D1(?) is not even a closed operator.

Hieber, Matthias and Wood, Ian (2007) The Dirichlet problem in convex bounded domains for operators in non-divergence form with L?-coefficients. Differential Integral Equations, 20 (7). pp. 721-734. ISSN 0893-4983.

Abstract

Consider the Dirichlet problem for elliptic and parabolic equations in nondivergence form with variable coefficients in convex bounded domains of $\R^n$. We prove solvability of the elliptic problem and maximal $L^q$-$L^p$-estimates for the solution of the parabolic problem provided the coefficients $a_{ij} \in L^\infty$ satisfy a Cordes condition and $p \in (1,2]$ is close to $2$. This implies that in two dimensions, i.e. $n=2$, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and $p \in (1,2]$ is close to $2$, for maximal $L^q$-$L^p$-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.

Geissert, Matthias and Heck, Horst and Hieber, Matthias et al. (2005) The Ornstein-Uhlenbeck semigroup in exterior domains. Archiv der Mathematik, 85 (6). pp. 554-562. ISSN 0003-889X.

Abstract

Let Omega be an exterior domain in R^n. It is shown that Ornstein-Uhlenbeck operators L generate C_0-semigroups on L^p(Omega) for p in (1, \infty) provided Omega is smooth. The method presented also allows to determine the domain D(L) of L and to prove L^p-L^q smoothing properties of e^{tL}. If Omega is only Lipschitz, results of this type are shown to be true for p close to 2.

Book Sections

Brown, Brian Malcolm and Evans, William Desmond and Wood, Ian (2013) Some spectral properties of Rooms and Passages domains and their skeletons. In: Holden, Helge and Simon, Barry and Teschl, Gerald Spectral analysis, differential equations and mathematical physics: a festschrift in honor of Fritz Gesztesy's 60th birthday. Proceedings of Symposia in Pure Mathematics, 87. AMS, Providence, RI, USA, pp. 69-85. ISBN 978-0-8218-7574-2.

Abstract

In this paper we investigate spectral properties of Laplacians on Rooms and Passages domains. In the first part, we use Dirichlet-Neumann bracketing techniques to show that for the Neumann Laplacian in certain Rooms and Passages domains the second term of the asymptotic expansion of the counting function is of order $\sqrt{\lambda}$. For the Dirichlet Laplacian our methods only give an upper estimate of the form $\sqrt{\lambda}$. In the second part of the paper, we consider the relationship between Neumann Laplacians on Rooms and Passages domains and Sturm-Liouville operators on the skeleton.

Brown, Brian Malcolm and Hoang, Vu and Plum, Michael et al. (2011) Floquet-Bloch Theory for Elliptic Problems with Discontinuous Coefficients. In: Janas, Jan and Kurasov, Pavel and Laptev, Ari et al. Spectral Theory and Analysis: Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP) 2008. Operator Theory: Advances and Applications. Springer, Poland, pp. 1-20. ISBN 9783764399931.

Abstract

We study spectral properties of elliptic problems of order 2m with periodic coefficients in L?. Our goal is to obtain a Floquet-Bloch type representation of the spectrum in terms of the spectra of associated operators acting on the period cell. Our approach using bilinear forms and operators in H?m-type spaces easily handles discontinuous coefficients and has the merit of being rather direct. In addition, the cell of periodicity is allowed to be unbounded, i.e. periodicity is not required in all spatial directions.

Brown, Brian Malcolm and Hoang, Vu and Plum, Michael et al. (2009) On Spectral Bounds for Photonic Crystal Waveguides. In: Bandle, Catherine and Losonczi, Laszlo and Gilanyi, Attila et al. Inequalities and Applications. International Series of Numerical Mathematics, 157. Birkhaeuser, Basel, pp. 23-30. ISBN 9783764387723.

Abstract

For a (d + 1)-dimensional photonic crystal with a linear defect strip (waveguide), we calculate real intervals containing spectrum of the associated spectral problem. If such an interval falls completely into a spectral gap of the unperturbed problem (without defect), this will prove the existence of additional spectrum induced by the waveguide.

Wood, Ian (2009) The Ornstein-Uhlenbeck Semigroup in Bounded and Exterior Lipschitz Domains. In: Janas, Jan and Kurasov, Pavel and Naboko, Serguei et al. Methods of Spectral Analysis in Mathematical Physics. Operator Theory: Advances and Applications, 186. Birkhaeuser, Basel, pp. 415-435. ISBN 9783764387549.

Abstract

We consider bounded Lipschitz domains ? in ? n . It is shown that the Dirichlet-Laplacian generates an analytic C 0-semigroup on L p (?) for p in an interval around 2 and that the corresponding Cauchy problem has the maximal L q -regularity property. We then prove that for bounded or exterior Lipschitz domains Ornstein-Uhlenbeck operators A generate C 0-semigroups in the same p-interval. The method, also allows to determine the domain D(A) of A and, if ? satisfies an outer ball condition, allows to show L p -L q -smoothing properties of the semigroups.

Hieber, Matthias and Wood, Ian (2003) Asymptotics of perturbations to the wave equation. In: UNSPECIFIED Evolution equations. Lecture Notes in Pure and Applied Mathematics, 234. Marcel Dekker, New York, pp. 243-252.

Edited Books

Brown, Brian Malcolm and Lang, Jan and Wood, Ian (2012) Spectral Theory, Function Spaces and Inequalities. Operator Theory: Advances and Applications, 219. Birkhaeuser, Basel, 264 pp. ISBN 978-3-0348-0262-8.

Abstract

This is a collection of contributed papers which focus on recent results in areas of differential equations, function spaces, operator theory and interpolation theory. In particular, it covers current work on measures of non-compactness and real interpolation, sharp Hardy-Littlewood-Sobolev inequalites, the HELP inequality, error estimates and spectral theory of elliptic operators, pseudo differential operators with discontinuous symbols, variable exponent spaces and entropy numbers. These papers contribute to areas of analysis which have been and continue to be heavily influenced by the leading British analysts David Edmunds and Des Evans. This book marks their respective 80th and 70th birthdays.

Conference Items
Total publications in KAR: 20 [See all in KAR]