Sharp Strichartz estimates for some variable coefficient Schrödinger operators on $ \mathbb{R}\times\mathbb{T}^2 $

<abstract><p>In the first part of the paper we continue the study of solutions to Schrödinger equations with a time singularity in the dispersive relation and in the periodic setting. In the second we show that if the Schrödinger operator involves a Laplace operator with variable coefficients with a particular dependence on the space variables, then one can prove Strichartz estimates at the same regularity as that needed for constant coefficients. Our work presents a two dimensional analysis, but we expect that with the obvious adjustments similar results are available in higher dimensions.</p></abstract>

Free and Forced Vibration Analysis of Peridynamic Finite Bar

Peridynamics is a reformulated nonlocal elasticity theory. Unlike the local elasticity theory, the peridynamics is proposed with no continuum assumption. In this paper, a new analytical approach to analyze the vibration of peridynamic finite bar with specified boundary condition is proposed. It is proved that the nonlocal dispersive relation of the peridynamic bar is nonlinear and can be reduced to the local dispersive relation when the peridynamic horizon goes to zero. The phase velocity, as a function of the wave frequency, is proved to be positive and asymptotically decreasing. The homogenous and the nonhomogeneous solutions of the peridynamic bar vibration equation are derived analytically by using the separation of variables. The mode shape characteristic equation of peridynamic bar, which is a second kind Fredholm integral equation, is expanded with a Taylor series expansion up to the infinite order; the corresponding mode shape is derived by solving a differential equation up to the infinite order. The peridynamic boundary condition is analyzed and compared with the local boundary condition. The numerical modeling based on mesh-free method verifies the analytical results for both free vibration and forced vibration cases.