Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Landesman-Lazer conditions at half-eigenvalues of the p-Laplacian

We study the existence of solutions of the Dirichlet problem {gather} -\phi_p(u')' -a_+ \phi_p(u^+) + a_- \phi_p(u^-) -\lambda \phi_p(u) = f(x,u), \quad x \in (0,1), \label{pb.eq} \tag{1} u(0)=u(1)=0,\label{pb_bc.eq} \tag{2} {gather} where $p>1$, \phi_p(s):=|s|^{p-1}\sgn s for $s \in \mathbb{R}$, the coefficients $a_\pm \in C^0[0,1]$, $\lambda \in \mathbb{R}$, and $u^\pm := \max\{\pm u,0\}$. We suppose that $f\in C^1([0,1]\times\mathbb{R})$ and that there exists $f_\pm \in C^0[0,1]$ such that $\lim_{\xi\to\pm\infty} f(x,\xi) = f_\pm(x)$, for all $x \in [0,1]$. With these conditions the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to have a `jumping nonlinearity'. We also suppose that the problem {gather} -\phi_p(u')' = a_+ \phi_p(u^+) - a_- \phi_p(u^-) + \lambda \phi_p(u) \quad\text{on} \ (0,1), \tag{3} \label{heval_pb.eq} {gather} together with \eqref{pb_bc.eq}, has a non-trivial solution $u$. That is, $\lambda$ is a `half-eigenvalue' of \eqref{pb_bc.eq}-\eqref{heval_pb.eq}, and the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem \eqref{pb.eq}-\eqref{pb_bc.eq} has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients $a_\pm$ are constants, and the resonance has been at a point in the `Fucik spectrum'. Even in this constant coefficient case our result extends previous results. In particular, previous variational approaches have required strong conditions on the location of the resonant point, whereas our result applies to any point in the Fucik spectrum.Comment: 14 page

Photonic monitoring of atmospheric fauna

Insects play a quintessential role in the Earth’s ecosystems and their recent decline in abundance and diversity is alarming. Monitoring their population is paramount to understand the causes of their decline, as well as to guide and evaluate the efficiency of conservation policies. Monitoring populations of flying insects is generally done using physical traps, but this method requires long and expensive laboratory analysis where each insect must be identified by qualified personnel. Lack of reliable data on insect populations is now considered a significant issue in the field of entomology, often referred to as a “data crisis” in the field. This doctoral work explores the potential of entomological photonic sensors to unlock some of the limitations of traditional methods. This work focuses on the development of optical instruments similar in essence to lidar systems, with the goal of counting and identifying flying insects from a distance in their natural habitat. Those systems rely on the interactions between the near-infrared laser light and insects flying through the laser beam. Each insect is characterized by retrieving its optical and morphological properties, such as wingbeat frequency, optical cross sections, or depolarization ratios. This project ran in parallel a series of laboratory and field experiments. In the laboratory, prototypes were tested and used to create a database of insects’ properties. The data were used to train machine learning classifiers aiming at identifying insects from optical signals. In the case of mosquitoes, the sex and species of an unknown specimen was predicted with a 99% and 80% accuracy respectively. It also showed that the presence of eggs within the abdomen of a female mosquito could be detected from several meters away with 87% accuracy. In the field, instruments were deployed in real-world conditions for a total of 520 days over three years. More than a million insects were observed, allowing to continuously monitor their aerial density over months with a temporal resolution down to the minute. While this approach remains very new, this work demonstrated that photonic sensors could become a powerful tool to tackle the current lack of data in the field of entomology

Second order, multi-point problems with variable coefficients

In this paper we consider the eigenvalue problem consisting of the equation -u" = \la r u, \quad \text{on $(-1,1)$}, where $r \in C^1[-1,1], \ r>0$ and \la \in \R, together with the multi-point boundary conditions u(\pm 1) = \sum^{m^\pm}_{i=1} \al^\pm_i u(\eta^\pm_i), where $m^\pm \ge 1$ are integers, and, for $i = 1,...,m^\pm$, \al_i^\pm \in \R, $\eta_i^\pm \in [-1,1]$, with $\eta_i^+ \ne 1$, $\eta_i^- \ne -1$. We show that if the coefficients \al_i^\pm \in \R are sufficiently small (depending on $r$) then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients \al_i^\pm are not sufficiently small then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case ($r \equiv 1$), but the variable coefficient case has not been considered previously (apart from the existence of `principal' eigenvalues). Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for existence of general solutions and also of nodal solutions --- these results rely on the spectral properties of the linear problem

High power femtosecond source based on passively mode-locked 1055nm VECSEL and Yb-fibre power amplifier

We report 5 ns pulses at 160 W average power and 910 repetition rate from a passively mode-locked VECSEL source seeding an Yb-doped fibre power amplifier. The amplified pulses were compressed to 291 fs duration

2s exciton-polariton revealed in an external magnetic field

We demonstrate the existence of the excited state of an exciton-polariton in a semiconductor microcavity. The strong coupling of the quantum well heavy-hole exciton in an excited 2s state to the cavity photon is observed in non-zero magnetic field due to surprisingly fast increase of Rabi energy of the 2s exciton-polariton in magnetic field. This effect is explained by a strong modification of the wave-function of the relative electron-hole motion for the 2s exciton state.Comment: 5 pages, 5 figure

Cancellation of probe effects in measurements of spin polarized momentum density by electron positron annihilation

Measurements of the two dimensional angular correlation of the electron-positron annihilation radiation have been done in the past to detect the momentum spin density and the Fermi surface. We point out that the momentum spin density and the Fermi Surface of ferromagnetic metals can be revealed within great detail owing to the large cancellation of the electron-positron matrix elements which in paramagnetic multiatomic systems plague the interpretation of the experiments. We prove our conjecture by calculating the momentum spin density and the Fermi surface of the half metal CrO2, who has received large attention due to its possible applications as spintronics material

Laser-driven plasma waves in capillary tubes

The excitation of plasma waves over a length of up to 8 centimeters is, for the first time, demon- strated using laser guiding of intense laser pulses through hydrogen filled glass capillary tubes. The plasma waves are diagnosed by spectral analysis of the transmitted laser radiation. The dependence of the spectral redshift, measured as a function of filling pressure, capillary tube length and incident laser energy, is in excellent agreement with simulation results. The longitudinal accelerating field inferred from the simulations is in the range 1 -10 GV/m