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)=-∫ 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 simpl...

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Autor principal: Rossi, Julio Daniel
Publicado: 2012
Materias:
Eigenvalues
Nonlocal diffusion
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v252_n12_p6429_Ignat
http://hdl.handle.net/20.500.12110/paper_00220396_v252_n12_p6429_Ignat
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_00220396_v252_n12_p6429_Ignat
record_format dspace
spelling paper:paper_00220396_v252_n12_p6429_Ignat2023-06-08T14:45:11Z Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space Rossi, Julio Daniel Eigenvalues Nonlocal diffusion 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. Fil:Rossi, J.D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. 2012 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v252_n12_p6429_Ignat http://hdl.handle.net/20.500.12110/paper_00220396_v252_n12_p6429_Ignat
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Eigenvalues
Nonlocal diffusion
spellingShingle Eigenvalues
Nonlocal diffusion
Rossi, Julio Daniel
Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
topic_facet Eigenvalues
Nonlocal diffusion
description 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.
author Rossi, Julio Daniel
author_facet Rossi, Julio Daniel
author_sort Rossi, Julio Daniel
title Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
title_short Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
title_full Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
title_fullStr Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
title_full_unstemmed Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
title_sort lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
publishDate 2012
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00220396_v252_n12_p6429_Ignat
http://hdl.handle.net/20.500.12110/paper_00220396_v252_n12_p6429_Ignat
work_keys_str_mv AT rossijuliodaniel lowerandupperboundsforthefirsteigenvalueofnonlocaldiffusionproblemsinthewholespace
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