218 research outputs found

    Wavelet frame bijectivity on Lebesgue and Hardy spaces

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    We prove a sufficient condition for frame-type wavelet series in LpL^p, the Hardy space H1H^1, and BMO. For example, functions in these spaces are shown to have expansions in terms of the Mexican hat wavelet, thus giving a strong answer to an old question of Meyer. Bijectivity of the wavelet frame operator acting on Hardy space is established with the help of new frequency-domain estimates on the Calder\'on-Zygmund constants of the frame kernel.Comment: 23 pages, 7 figure

    Magnetic spectral bounds on starlike plane domains

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    We develop sharp upper bounds for energy levels of the magnetic Laplacian on starlike plane domains, under either Dirichlet or Neumann boundary conditions and assuming a constant magnetic field in the transverse direction. Our main result says that ∑j=1nΦ(λjA/G)\sum_{j=1}^n \Phi \big( \lambda_j A/G \big) is maximal for a disk whenever Φ\Phi is concave increasing, n≥1n \geq 1, the domain has area AA, and λj\lambda_j is the jj-th Dirichlet eigenvalue of the magnetic Laplacian (i∇+β2A(−x2,x1))2\big( i\nabla+ \frac{\beta}{2A}(-x_2,x_1) \big)^2. Here the flux β\beta is constant, and the scale invariant factor GG penalizes deviations from roundness, meaning G≥1G \geq 1 for all domains and G=1G=1 for disks

    Nonlinear dynamics of phase separation in thin films

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    We present a long-wavelength approximation to the Navier-Stokes Cahn-Hilliard equations to describe phase separation in thin films. The equations we derive underscore the coupled behaviour of free-surface variations and phase separation. We introduce a repulsive substrate-film interaction potential and analyse the resulting fourth-order equations by constructing a Lyapunov functional, which, combined with the regularizing repulsive potential, gives rise to a positive lower bound for the free-surface height. The value of this lower bound depends on the parameters of the problem, a result which we compare with numerical simulations. While the theoretical lower bound is an obstacle to the rupture of a film that initially is everywhere of finite height, it is not sufficiently sharp to represent accurately the parametric dependence of the observed dips or `valleys' in free-surface height. We observe these valleys across zones where the concentration of the binary mixture changes sharply, indicating the formation of bubbles. Finally, we carry out numerical simulations without the repulsive interaction, and find that the film ruptures in finite time, while the gradient of the Cahn--Hilliard concentration develops a singularity.Comment: 26 pages, 20 figures, PDFLaTeX with RevTeX4 macros. A thorough analysis of the equations is presented in arXiv:0805.103

    Maximizing Neumann fundamental tones of triangles

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    We prove sharp isoperimetric inequalities for Neumann eigenvalues of the Laplacian on triangular domains. The first nonzero Neumann eigenvalue is shown to be maximal for the equilateral triangle among all triangles of given perimeter, and hence among all triangles of given area. Similar results are proved for the harmonic and arithmetic means of the first two nonzero eigenvalues
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