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Abstract:

We find lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form T(u)=-∫ ℝdK(x,y)(u(y)-u(x))dy. Here we consider a kernel K(x, y)=ψ(y-a(x))+ψ(x-a(y)) where ψ is a bounded, nonnegative function supported in the unit ball and a means a diffeomorphism on ℝ 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 ψ. Indeed, in the linear case a(x)=Ax we obtain an explicit expression for the first eigenvalue in the whole ℝ d and it is positive when 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 behavior 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→∞. © 2012 Elsevier Inc.

Registro:

Documento: Artículo
Título:Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
Autor:Ignat, L.I.; Rossi, J.D.; San Antolin, A.
Filiación:Inst. of Mathematics Simion Stoilow of the Romanian Academy, 21 Calea Grivitei Street, 010702 Bucharest, Romania
BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009 Bilbao, Basque-Country, Spain
Departamento de Análisis Matemático, Universidad de Alicante, Ap. correos 99, 03080 Alicante, Spain
Dpto. de Matemáticas, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
Palabras clave:Eigenvalues; Nonlocal diffusion
Año:2012
Volumen:252
Número:12
Página de inicio:6429
Página de fin:6447
DOI: http://dx.doi.org/10.1016/j.jde.2012.03.011
Título revista:Journal of Differential Equations
Título revista abreviado:J. Differ. Equ.
ISSN:00220396
CODEN:JDEQA
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v252_n12_p6429_Ignat

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Citas:

---------- APA ----------
Ignat, L.I., Rossi, J.D. & San Antolin, A. (2012) . Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space. Journal of Differential Equations, 252(12), 6429-6447.
http://dx.doi.org/10.1016/j.jde.2012.03.011
---------- CHICAGO ----------
Ignat, L.I., Rossi, J.D., San Antolin, A. "Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space" . Journal of Differential Equations 252, no. 12 (2012) : 6429-6447.
http://dx.doi.org/10.1016/j.jde.2012.03.011
---------- MLA ----------
Ignat, L.I., Rossi, J.D., San Antolin, A. "Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space" . Journal of Differential Equations, vol. 252, no. 12, 2012, pp. 6429-6447.
http://dx.doi.org/10.1016/j.jde.2012.03.011
---------- VANCOUVER ----------
Ignat, L.I., Rossi, J.D., San Antolin, A. Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space. J. Differ. Equ. 2012;252(12):6429-6447.
http://dx.doi.org/10.1016/j.jde.2012.03.011