1,024 research outputs found
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 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, a nonnegative kernel and
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 of the problem in the space of signed
measure. Moreover a positive measure. When is absolutely continuous
with respect to the Lebesgue measure, is called the principal
eigenfunction associated to . 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, a nonnegative
kernel and a positive function. First, we establish a criterion for the
existence of a principal eigenpair . 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 , and prove the existence of
waves when 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
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