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26079 results in Abstract analysis

Page 393 of 1304

  • Waveless subcritical flow past symmetric bottom topography

  • R. J. HOLMES, G. C. HOCKING, L. K. FORBES, N. Y. BAILLARD
  • Journal:
    European Journal of Applied Mathematics / Volume 24 / Issue 2 / April 2013
    Published online by Cambridge University Press:
    09 November 2012, pp. 213-230

  • The subcritical flow of a stream over a bottom obstruction or depression is considered with particular interest in obtaining solutions with no downstream waves. In the linearised problem this can always be achieved by superposition of multiple obstructions, but it is not clear whether this is possible in a full nonlinear problem. Solutions computed here indicate that there is an effective nonlinear superposition principle at work as no special shape modifications were required to obtain wave-cancelling solutions. Waveless solutions corresponding to one or more trapped waves are computed at a range of different Froude numbers and are shown to provide a rather elaborate mosaic of solution curves in parameter space when both negative and positive obstruction heights are included.

  • Existence results for diffuse interface models describing phase separation and damage*

  • CHRISTIAN HEINEMANN, CHRISTIANE KRAUS
  • Journal:
    European Journal of Applied Mathematics / Volume 24 / Issue 2 / April 2013
    Published online by Cambridge University Press:
    09 November 2012, pp. 179-211

  • In this paper, we analytically investigate multi-component Cahn–Hilliard and Allen–Cahn systems which are coupled with elasticity and uni-directional damage processes. The free energy of the system is of the form ∫Ω½Γ∇c : ∇c + ½|∇z|2+Wch(c)+Wel(e,c,z)dx with a polynomial or logarithmic chemical energy density Wch, an inhomogeneous elastic energy density Wel and a quadratic structure of the gradient of damage variable z. For the corresponding elastic Cahn–Hilliard and Allen–Cahn systems coupled with uni-directional damage processes, we present an appropriate notion of weak solutions and prove existence results based on certain regularization methods and a higher integrability result for strain e.

Topics in Critical Point Theory
  • Kanishka Perera, Martin Schechter
  • Published online:
    05 November 2012
    Print publication:
    01 November 2012
  • This book introduces the reader to powerful methods of critical point theory and details successful contemporary approaches to many problems, some of which had proved resistant to attack by older methods. Topics covered include Morse theory, critical groups, the minimax principle, various notions of linking, jumping nonlinearities and the Fučík spectrum in an abstract setting, sandwich pairs and the cohomological index. Applications to semilinear elliptic boundary value problems, p-Laplacian problems and anisotropic systems are given. Written for graduate students and research scientists, the book includes numerous examples and presents more recent developments in the subject to bring the reader up to date with the latest research.

Operator Methods for Boundary Value Problems
  • Edited by Seppo Hassi, Hendrik S. V. de Snoo, Franciszek Hugon Szafraniec
  • Published online:
    05 November 2012
    Print publication:
    11 October 2012
  • Presented in this volume are a number of new results concerning the extension theory and spectral theory of unbounded operators using the recent notions of boundary triplets and boundary relations. This approach relies on linear single-valued and multi-valued maps, isometric in a Krein space sense, and offers a basic framework for recent developments in system theory. Central to the theory are analytic tools such as Weyl functions, including Titchmarsh-Weyl m-functions and Dirichlet-to-Neumann maps. A wide range of topics is considered in this context from the abstract to the applied, including boundary value problems for ordinary and partial differential equations; infinite-dimensional perturbations; local point-interactions; boundary and passive control state/signal systems; extension theory of accretive, sectorial and symmetric operators; and Calkin's abstract boundary conditions. This accessible treatment of recent developments, written by leading researchers, will appeal to a broad range of researchers, students and professionals.

  • Preface

  • Kanishka Perera, Florida Institute of Technology, Martin Schechter, University of California, Irvine
  • Book:
    Topics in Critical Point Theory
    Published online:
    05 November 2012
    Print publication:
    01 November 2012, pp ix-xii

  • Summary

    Critical point theory has become a very powerful tool for solving many problems. The theory has enjoyed significant development over the past several years. The impetus for this development is the fact that many new problems could not be solved by the older theory.

    There have been several excellent books written on critical point theory from various points of view; see, e.g., Berger [19], Zeidler [161], Rabinowitz [129], Mawhin and Willem [91], Chang [29, 30], Ghoussoub [56], Ambrosetti and Prodi [8], Willem [158], Chabrowski [26], Dacorogna [36], and Struwe [153] (see also Schechter [143, 144, 147], Zou and Schechter [163], and Perera et al. [113]). In this book we present more recent developments in the subject that do not seem to be covered elsewhere, including some results of the authors dealing with nonstandard linking geometries and sandwich pairs.

    Chapter 1 is a brief review of Morse theory in Banach spaces. We prove the first and second deformation lemmas under the Cerami compactness condition. As the variational functionals associated with applications given later in the book will only be C1, we discuss critical groups of C1-functionals. We include discussions on minimizers, nontrivial critical points, mountain pass points, and the three critical points theorem. We also give a generalized notion of local linking that yields a nontrivial critical group, which will be applied to problems with jumping nonlinearities in Chapter 5. We close the chapter with a recent result of Perera [110] on nontrivial critical groups in p-Laplacian problems.

  • EJM volume 23 issue 6 Cover and Front matter

  • Journal:
    European Journal of Applied Mathematics / Volume 23 / Issue 6 / December 2012
    Published online by Cambridge University Press:
    30 October 2012, pp. f1-f2
    • Article
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  • Sums of Dilates in p

  • GONZALO FIZ PONTIVEROS
  • Journal:
    Combinatorics, Probability and Computing / Volume 22 / Issue 2 / March 2013
    Published online by Cambridge University Press:
    30 October 2012, pp. 282-293

  • We consider the problem of sums of dilates in groups of prime order. It is well known that sets with small density and small sumset in p behave like integer sets. Thus, given A ⊂ p of sufficiently small density, it is straightforward to show that

    \begin{linenomath}$$| \lambda_{1}A+\lambda_{2}A+\cdots+ \lambda_{k}A | \ge\biggl(\sum_{i}|\lambda_{i}|\biggr)|A|- o(|A|).$$\end{linenomath}
    On the other hand, the behaviour for sets of large density turns out to be rather surprising. Indeed, for any ε > 0, we construct subsets of density 1/2–ε such that |A + λ A| ≤ (1–δ)p, showing that there is a very different behaviour for subsets of large density.

  • EJM volume 23 issue 6 Cover and Back matter

  • Journal:
    European Journal of Applied Mathematics / Volume 23 / Issue 6 / December 2012
    Published online by Cambridge University Press:
    30 October 2012, pp. b1-b8
    • Article
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  • A model for the evolution of the thermal bar system

  • DUNCAN E. FARROW
  • Journal:
    European Journal of Applied Mathematics / Volume 24 / Issue 2 / April 2013
    Published online by Cambridge University Press:
    30 October 2012, pp. 161-177

  • A new framework for modelling the evolution of the thermal bar system in a lake is presented. The model assumes that the thermal bar is located between two regions: the deeper region, where spring warming leads to overturning of the entire water column, and the near shore shallower region, where a stable surface layer is established. In this model the thermal bar moves out slightly more quickly than predicted by a simple thermal balance. Also, the horizontal extent of the thermal bar region increases as it moves out from the shore.

Page 393 of 1304