Vol 226, No 4 (2017)

We present a simple proof of the boundedness of holomorphic and harmonic Bergman projection operators on a half-plane and a half-space respectively on the Orlicz space, the variable exponent Lebesgue space, and the variable exponent generalized Morrey space. The approach is based on an idea due to V. P. Zaharyuta and V. I. Yudovich (1962) to use Calderón–Zygmund operators for proving the boundedness of the Bergman projection in Lebesgue spaces on the unit disc. We also study the rate of growth of functions near the boundary in the spaces under consideration.

We obtain the asymptotically exact weighted Korn inequality for a thin (of thickness δh) periodically corrugated (with step δ) beam clamped at its ends. The constant in the inequality is independent of both small parameters δ and h, but the distribution of weighted factors under longitudinal and transverse displacements essentially differs from the optimal one in the case of a straight beam.

We find asymptotic representations of eigenvalues inside gaps of the continuous spectrum of a periodic waveguide with local smooth gently sloped (of depth ε ≪ 1) perturbations of walls. These eigenvalues reach the upper or lower gap edge as ε → +0. We consider several variants of the gap edge structure and obtain conditions guaranteeing the existence or absence of points of the discrete spectrum in small neighborhoods. We calculate the total number of eigenvalues in a gap for small ε. To justify the asymptotic expansions, we use elementary tools of the theory of spectral measure.

We study the nonlinear eigenvalue problem for two-dimensional Hartree type equations with selfaction potentials possessing logarithmic singularity and depending on the distance between points. To find a series of asymptotic eigenvalues, we derive a counterpart of the Bohr–Sommerfeld quantization rule. The corresponding asymptotic eigenfunctions are localized near a plane segment.

We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semiclassical parameter. We show that the ground state energy of the Pauli operator is exponentially small as the semiclassical parameter tends to zero and estimate the decay rate. This extends our recent results discussing a recent paper by Ekholm–Kovařík–Portmann, including non-simply connected domains.