2015/12/21 by Henri Berestycki, Berestycki, Henri, Jérôme Coville +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1512.06529
openalex publication_date 2015/12/21 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/29
In this article we study some spectral properties of the linear operator\n\L\_\Ω+a defined on the space C( Ω) by :\n
mathcalL
Omega[
varphi]\n+a
varphi:=
int
OmegaK(x,y)
varphi(y)
,dy+a(x)
varphi(x) where\n\Ω\⊂ \ℝN is a domain, possibly unbounded, a is a\ncontinuous bounded function and K is a continuous, non negative kernel\nsatisfying an integrability condition. We focus our analysis on the properties\nof the generalised principal eigenvalue \λ\_p(\L\_\Ω+a)\ndefined by
lambda
p(
mathcalL
Omega+a):=
sup
lambda
in
mathbbR\n
,|
,
exists
varphi
in C(
bar
Omega),
varphi
textgreater0,
textitsuch\nthat
,
mathcalL
Omega[
varphi] +a
varphi +
lambda
varphi
le 0
,\n
textin
;
Omega
. We establish some new properties of this generalised\nprincipal eigenvalue \λ\_p. Namely, we prove the equivalence of\ndifferent definitions of the principal eigenvalue. We also study the behaviour\nof \λ\_p(\L\_\Ω+a) with respect to some scaling of K.\nFor kernels K of the type, K(x,y)=J(x-y) with J a compactly supported\nprobability density, we also establish some asymptotic properties of\n\λ\_p \(\L\_\σ,m,\Ω\n-\(1)/(\σm)+a\) where \L\_\σ,m,\Ω is defined\nby\n\\L\_\σ,m,\Ω[\φ]:= frac1\σ2+N\∫\_\ΩJ\(\(x-y)/(\σ)\)\φ(y) ,\ndy. In particular, we prove that
lim
sigma
to\n0
lambda
p
left(
mathcalL
sigma,2,
Omega-
frac1
sigma2+a
right)=
lambda
1
left(
fracD
2(J)2N
Delta\n+a
right),where D\_2(J):=\∫\_\ℝNJ(z)|z|2 ,dz and \λ\_1\ndenotes the Dirichlet principal eigenvalue of the elliptic operator. In\naddition, we obtain some convergence results for the corresponding\neigenfunction \φ\_p,\σ.\n