Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lower and upper bounds for the first eigenvalue of nonlocal diffusion
  problems in the whole space   We find lower and upper bounds for the first eigenvalue of a nonlocal
diffusion operator of the form $ T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \,
dy$. Here we consider a kernel $K(x,y)=\psi (y-a(x))+\psi(x-a(y))$ where $\psi$
is a bounded, nonnegative function supported in the unit ball and $a$ means a
diffeomorphism on $\rr^d$. A simple example being a linear function $a(x)= Ax$.
The upper and lower bounds that we obtain are given in terms of the Jacobian of
$a$ and the integral of $\psi$. Indeed, in the linear case $a(x) = Ax$ we
obtain an explicit expression for the first eigenvalue in the whole $\rr^d$ and
it is positive when the the determinant of the matrix $A$ is different from
one. As an application of our results, we observe that, when the first
eigenvalue is positive, there is an exponential decay for the solutions to the
associated evolution problem. As a tool to obtain the result, we also study the
behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the
ball $B_R$ and prove that it converges to the first eigenvalue in the whole
space as $R\to \infty$.

22
The categories of this paper are: chao-dyn
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
0
The categories of this paper are: hep-ph
19
The categories of this paper are: eess
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
34
The categories of this paper are: chem-ph
1
The categories of this paper are: cs
27
The categories of this paper are: comp-gas
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The categories of this paper are: stat
32
The categories of this paper are: atom-ph
26
The categories of this paper are: bayes-an
36
The categories of this paper are: cmp-lg