6,436 research outputs found
Best dispersal strategies in spatially heterogeneous environments: optimization of the principal eigenvalue for indefinite fractional Neumann problems
We study the positive principal eigenvalue of a weighted problem associated
with the Neumann spectral fractional Laplacian. This analysis is related to the
investigation of the survival threshold in population dynamics. Our main result
concerns the optimization of such threshold with respect to the fractional
order , the case corresponding to the standard Neumann
Laplacian: when the habitat is not too fragmented, the principal positive
eigenvalue can not have local minima for . As a consequence, the best
strategy for survival is either following the diffusion with (i.e.
Brownian diffusion), or with the lowest possible (i.e. diffusion allowing
long jumps), depending on the size of the domain. In addition, we show that
analogous results hold for the standard fractional Laplacian in ,
in periodic environments.Comment: Version accepted for publication. Title changed according to
referee's suggestio
Dynamic isoperimetry and the geometry of Lagrangian coherent structures
The study of transport and mixing processes in dynamical systems is
particularly important for the analysis of mathematical models of physical
systems. We propose a novel, direct geometric method to identify subsets of
phase space that remain strongly coherent over a finite time duration. This new
method is based on a dynamic extension of classical (static) isoperimetric
problems; the latter are concerned with identifying submanifolds with the
smallest boundary size relative to their volume.
The present work introduces \emph{dynamic} isoperimetric problems; the study
of sets with small boundary size relative to volume \emph{as they are evolved
by a general dynamical system}. We formulate and prove dynamic versions of the
fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming
theorem and a dynamic Cheeger inequality. We introduce a new dynamic Laplacian
operator and describe a computational method to identify coherent sets based on
eigenfunctions of the dynamic Laplacian.
Our results include formal mathematical statements concerning geometric
properties of finite-time coherent sets, whose boundaries can be regarded as
Lagrangian coherent structures. The computational advantages of our new
approach are a well-separated spectrum for the dynamic Laplacian, and
flexibility in appropriate numerical approximation methods. Finally, we
demonstrate that the dynamic Laplacian operator can be realised as a
zero-diffusion limit of a newly advanced probabilistic transfer operator method
(Froyland, 2013) for finding coherent sets, which is based on small diffusion.
Thus, the present approach sits naturally alongside the probabilistic approach
(Froyland, 2013), and adds a formal geometric interpretation
The asymptotic behaviour of the heat equation in a twisted Dirichlet-Neumann waveguide
We consider the heat equation in a straight strip, subject to a combination
of Dirichlet and Neumann boundary conditions. We show that a switch of the
respective boundary conditions leads to an improvement of the decay rate of the
heat semigroup of the order of . The proof employs similarity
variables that lead to a non-autonomous parabolic equation in a thin strip
contracting to the real line, that can be analyzed on weighted Sobolev spaces
in which the operators under consideration have discrete spectra. A careful
analysis of its asymptotic behaviour shows that an added Dirichlet boundary
condition emerges asymptotically at the switching point, breaking the real line
in two half-lines, which leads asymptotically to the 1/2 gain on the spectral
lower bound, and the gain on the decay rate in the original physical
variables.
This result is an adaptation to the case of strips with twisted boundary
conditions of previous results by the authors on geometrically twisted
Dirichlet tubes.Comment: 15 pages, 2 figure
Multiple positive solutions for a class of p-Laplacian Neumann problems without growth conditions
For , we consider the following problem where
is either a ball or an annulus. The nonlinearity
is possibly supercritical in the sense of Sobolev embeddings; in particular our
assumptions allow to include the prototype nonlinearity
for every . We use the shooting method to get existence and multiplicity
of non-constant radial solutions. With the same technique, we also detect the
oscillatory behavior of the solutions around the constant solution .
In particular, we prove a conjecture proposed in [D. Bonheure, B. Noris, T.
Weth, {\it Ann. Inst. H. Poincar\'e Anal. Non Lin\'aire} vol. 29, pp. 573-588
(2012)], that is to say, if and , there exists
a radial solution of the problem having exactly intersections with
for a large class of nonlinearities.Comment: 22 pages, 4 figure
Spectral problem on graphs and L-functions
The scattering process on multiloop infinite p+1-valent graphs (generalized
trees) is studied. These graphs are discrete spaces being quotients of the
uniform tree over free acting discrete subgroups of the projective group
. As the homogeneous spaces, they are, in fact, identical to
p-adic multiloop surfaces. The Ihara-Selberg L-function is associated with the
finite subgraph-the reduced graph containing all loops of the generalized tree.
We study the spectral problem on these graphs, for which we introduce the
notion of spherical functions-eigenfunctions of a discrete Laplace operator
acting on the graph. We define the S-matrix and prove its unitarity. We present
a proof of the Hashimoto-Bass theorem expressing L-function of any finite
(reduced) graph via determinant of a local operator acting on this
graph and relate the S-matrix determinant to this L-function thus obtaining the
analogue of the Selberg trace formula. The discrete spectrum points are also
determined and classified by the L-function. Numerous examples of L-function
calculations are presented.Comment: 39 pages, LaTeX, to appear in Russ. Math. Sur
A priori bounds and multiplicity of positive solutions for -Laplacian Neumann problems with sub-critical growth
Let and let be either a ball or an
annulus. We continue the analysis started in [Boscaggin, Colasuonno, Noris,
ESAIM Control Optim. Calc. Var. (2017)], concerning quasilinear Neumann
problems of the type -\Delta_p u = f(u), \quad u>0 \mbox{ in } \Omega, \quad
\partial_\nu u = 0 \mbox{ on } \partial\Omega. We suppose that
and that is negative between the two zeros and positive after. In case
is a ball, we also require that grows less than the
Sobolev-critical power at infinity. We prove a priori bounds of radial
solutions, focusing in particular on solutions which start above 1. As an
application, we use the shooting technique to get existence, multiplicity and
oscillatory behavior (around 1) of non-constant radial solutions.Comment: 26 pages, 3 figure
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