Singular measure as principal eigenfunction of some nonlocal operators

In this paper, we are interested in the spectral properties of the generalised principal eigenvalue of some nonlocal operator. That is, we look for the existence of some particular solution $(\lambda,\phi)$ of a nonlocal operator. \int_{\O}K(x,y)\phi(y)\, dy +a(x)\phi(x) =-\lambda \phi(x), where \O\subset\R^n is an open bounded connected set, $K$ a nonnegative kernel and $a$ is continuous. We prove that for the generalised principal eigenvalue \lambda_p:=\sup \{\lambda \in \R \, |\, \exists \, \phi \in C(\O), \phi > 0 \;\text{so that}\; \oplb{\phi}{\O}+ a(x)\phi + \lambda\phi\le 0\} there exists always a solution $(\mu, \lambda_p)$ of the problem in the space of signed measure. Moreover $\mu$ a positive measure. When $\mu$ is absolutely continuous with respect to the Lebesgue measure, $\mu =\phi_p(x)$ is called the principal eigenfunction associated to $\lambda_p$. In some simple cases, we exhibit some explicit singular measures that are solutions of the spectral problem

On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators

In this paper we are interested in the existence of a principal eigenfunction of a nonlocal operator which appears in the description of various phenomena ranging from population dynamics to micro-magnetism. More precisely, we study the following eigenvalue problem: \int_{\O}J(\frac{x-y}{g(y)})\frac{\phi(y)}{g^n(y)}\, dy +a(x)\phi =\rho \phi, where \O\subset\R^n is an open connected set, $J$ a nonnegative kernel and $g$ a positive function. First, we establish a criterion for the existence of a principal eigenpair $(\lambda_p,\phi_p)$. We also explore the relation between the sign of the largest element of the spectrum with a strong maximum property satisfied by the operator. As an application of these results we construct and characterize the solutions of some nonlinear nonlocal reaction diffusion equations

Nonlocal anisotropic dispersal with monostable nonlinearity

We study the travelling wave problem J\astu - u - cu' + f (u) = 0 in R, u(-\infty) = 0, u(+\infty) = 1 with an asymmetric kernel J and a monostable nonlinearity. We prove the existence of a minimal speed, and under certain hypothesis the uniqueness of the profile for c = 0. For c = 0 we show examples of nonuniqueness

Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq

Promoting Handwashing and Sanitation: Evidence From a Large-Scale Randomized Trial in Rural Tanzania

This paper presents the results of two large-scale, government-led handwashing and sanitation promotion campaigns in rural Tanzania. Their results highlight the importance of focusing on intermediate outcomes of take-up and behavior change as a critical first step in large-scale programs before realizing the changes in health that sanitation and hygiene interventions aim to deliver