Existence of nodal solutions for Dirac equations with singular nonlinearities

We prove, by a shooting method, the existence of infinitely many solutions of the form $\psi(x^0,x) = e^{-i\Omega x^0}\chi(x)$ of the nonlinear Dirac equation {equation*} i\underset{\mu=0}{\overset{3}{\sum}} \gamma^\mu \partial_\mu \psi- m\psi - F(\bar{\psi}\psi)\psi = 0 {equation*} where $\Omega>m>0,$ $\chi$ is compactly supported and \[F(x) = \{{array}{ll} p|x|^{p-1} & \text{if} |x|>0 0 & \text{if} x=0 {array}.] with $p\in(0,1),$ under some restrictions on the parameters $p$ and $\Omega.$ We study also the behavior of the solutions as $p$ tends to zero to establish the link between these equations and the M.I.T. bag model ones

Form removal aspects on the waviness parameters for steel sheet in automotive applications : fourier filtering versus polynomial regression

Premium car makers attach great importance to the visual appearance of the painted car skin as an indication of product quality. The “orange peel” phenomenon constitutes a major problem here. It is not only depending on the paint’s chemical composition and application method, but also on possible waviness components in the sheet substrate. Therefore one is searching hard for a valuable waviness parameter to quantify the substrate’s fitness for purpose. A technically emerging problem is how to remove the form from the measured signal, which is indeed not significant to the orange peel phenomenon. This paper will compare two commonly used approaches: i.e. Fourier filtering versus polynomial regression and will reveal and quantify some common aspects in terms of wavelengths

Estimates for a class of oscillatory integrals and decay rates for wave-type equations

This paper investigates higher order wave-type equations of the form $\partial_{tt}u+P(D_{x})u=0$, where the symbol $P(\xi)$ is a real, non-degenerate elliptic polynomial of the order $m\ge4$ on ${\bf R}^n$. Using methods from harmonic analysis, we first establish global pointwise time-space estimates for a class of oscillatory integrals that appear as the fundamental solutions to the Cauchy problem of such wave equations. These estimates are then used to establish (pointwise-in-time) $L^p-L^q$ estimates on the wave solution in terms of the initial conditions