Vol 242, No 2 (2019)

Let f be a positive homogeneous function of degree 0 defined on the sphere Σ of the space ℝⁿ, and let Φ_α be the symbol of the integral operator

\( \underset{{\mathrm{\mathbb{R}}}^n}{\int}\frac{f\left(\left(\mathrm{x}-y\right)/\left|\mathrm{x}-y\right|\right)}{{\left|\mathrm{x}-y\right|}^{n-\alpha }}u(y) dy,\kern1em u\in {C}_0^{\infty}\left({\mathrm{\mathbb{R}}}^n\right), \)

with 0 < α < n. We study differentiability properties of the restriction of Φ_α to the unit sphere Σ in the spaces \( {H}_p^l\left(\Sigma \right) \) for p ∈ (1,∞), where \( {H}_p^l\left(\Sigma \right) \) denotes the space of Bessel potentials with the norm \( {\left\Vert f\right\Vert}_{H_p^l\left(\Sigma \right)}={\left\Vert {\left(\delta +I\right)}^{l/2}f\right\Vert}_{L_p\left(\Sigma \right)} \) and δ is the Beltrami operator on the sphere. We prove that if f ∈ L_p(Σ), then \( {\left.{\Phi}_{\alpha}\right|}_{\Sigma}\in {H}_p^l\left(\Sigma \right) \) for any l ≤ n/2 − α − |p⁻¹ − 2⁻¹|(n − 2). Conversely, if \( {\left.{\Phi}_{\alpha}\right|}_{\Sigma}\in {H}_p^l\left(\Sigma \right) \) with |l ≥ n/2 − α+|p⁻¹ − 2⁻¹|(n − 2), then f ∈ L_p(Σ). The results are sharp.

We study the spectrum of a thin (with the relative width h ≪ 1) rectangular lattice of elastic isotropic (with the Lamé constants ⋋ ≥ 0 and μ > 0) plane waveguides simulating joining seams of a doubly periodic system of identical absolutely rigid tiles. We establish that the low-frequency range of the essential spectrum contains two spectral bands (passing ones for waves) of length O(e^−δ/(2h)), δ > 0. Above these bands there is a gap of width O(h⁻²) (stopping zones for waves) and then, in the mid-frequency range, above the cut-off value μπ²h⁻² of the continuous spectrum of the infinite cross-shaped waveguide, there is a family of spectral bands of length O(h); moreover, between some of these bands there are opened up gaps of width O(1). The character of the wave propagation depends on whether the frequencies are below or above the cut-off value. In the first case, the oscillations are strictly concentrated near the lattice nodes and the edges are practically immovable. In the second case, the oscillations are localized on the lattice edges, i.e., the nodes are left at relative rest. We show that single perturbations of nodes or edges can cause the appearance of points of the discrete spectrum under the essential spectrum or inside the gaps; moreover, an infinite collection of identical perturbations of nodes can also change the essential spectrum. Bibliography: 78 titles. Illustrations: 5 figures.

We refine the estimate for the phase transition temperatures in the variational problem on the equilibrium in a two-phase elastic medium, which makes it possible to obtain a coincidence criterion for the phase transition temperatures.

The TV-regularization method due to Rudin, Osher, and Fatemi is widely used in mathematical image analysis. We consider a nonstationary and iterative variant of this approach and provide a mathematical theory that extends the results of Radmoser et al. to the BV setting. While existence and uniqueness, a maximum–minimum principle, and preservation of the average grey value are not hard to prove, we also establish the convergence to a constant steady state and consider a large family of Lyapunov functionals. These properties allow us to interpret the iterated TV-regularization as a time-discrete scale-space representation of the original image.