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On the definition and the properties of the principal eigenvalue of some nonlocal operators

Abstract : In this article we study some spectral properties of the linear operator L-Omega + a defined on the space C((Omega) over bar) by: L-Omega[phi] + a phi := integral(Omega) K(x, y)phi(y) dy + a(x)phi(x) where Omega C R-N is a domain, possibly unbounded, a is a continuous bounded function and K is a continuous, non-negative kernel satisfying an integrability condition. We focus our analysis on the properties of the generalised principal eigenvalue lambda(p)(L-Omega + a) defined by lambda(p)(L-Omega + a) := sup{lambda is an element of R vertical bar there exists phi is an element of C((Omega) over bar), phi > 0, such that L-Omega[phi] + a phi + lambda phi <= 0 in Omega}. For kernels K of the type, K(x,y) = J(x - y) with J a compactly supported probability density, we also establish some asymptotic properties of lambda(p) (L-sigma,L-m,L-Omega - 1/sigma(m) +a) where L-sigma,L-m,L-Omega is defined by L-sigma,L-2,L-Omega[phi] := 1/sigma(2+N) integral J (x - y/sigma) phi(y) dy. In particular, we prove that lim(sigma -> 0) lambda(p) (L-sigma,L-m,L-Omega - 1/sigma(2) + a) = lambda(1) (D-2(J)/2N Delta+a), where D-2(J) := integral(RN) J(z)vertical bar z vertical bar(2) dz and lambda(1) enotes the Dirichlet principal eigenvalue of the elliptic operator. In addition, we obtain some convergence results for the corresponding eigenfunction phi(p,sigma).
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Submitted on : Thursday, May 28, 2020 - 5:14:20 PM
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Henri Berestycki, Jerome Coville, Hoang-Hung Vo. On the definition and the properties of the principal eigenvalue of some nonlocal operators. Journal of Functional Analysis, Elsevier, 2016, 271 (10), pp.2701 - 2751. ⟨10.1016/j.jfa.2016.05.017⟩. ⟨hal-02641531⟩



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