On the principal eigenvalue of some nonlocal diffusion problems

Abstract In this paper we analyze some properties of the principal eigenvalue λ 1 ( Ω ) of the nonlocal Dirichlet problem ( J ∗ u ) ( x ) − u ( x ) = − λ u ( x ) in Ω with u ( x ) = 0 in R N ∖ Ω . Here Ω is a smooth bounded domain of R N and the kernel J is assumed to be a C 1 compactly supported, even, nonnegative function with unit integral. Among other properties, we show that λ 1 ( Ω ) is continuous (or even differentiable) with respect to continuous (differentiable) perturbations of the domain Ω. We also provide an explicit formula for the derivative. Finally, we analyze the asymptotic behavior of the decreasing function Λ ( γ ) = λ 1 ( γ Ω ) when the dilatation parameter γ > 0 tends to zero or to infinity.

[1]  Julio D. Rossi,et al.  Maximum and antimaximum principles for some nonlocal diffusion operators , 2009 .

[2]  K. Schumacher.  Travelling-front solutions for integro-differential equations. I. , 1980 .

[3]  Adam J. J. Chmaj,et al.  Heteroclinic solutions of a van der Waals model with indefinite nonlocal interactions , 2005 .

[4]  M. Kreĭn,et al.  Linear operators leaving invariant a cone in a Banach space , 1950 .

[5]  J. Rossi,et al.  Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions , 2009 .

[6]  Jérôme Coville,et al.  On uniqueness and monotonicity of solutions of non-local reaction diffusion equation , 2006 .

[7]  Ademir Fernando Pazoto,et al.  Asymptotic behaviour for a semilinear nonlocal equation , 2007, Asymptot. Anal..

[8]  Noel G. Lloyd,et al.  CENTRES AND LIMIT CYCLES FOR AN EXTENDED KUKLES SYSTEM , 2007 .

[9]  Jérôme Coville,et al.  Propagation speed of travelling fronts in non local reaction–diffusion equations , 2005 .

[10]  Liviu I. Ignat,et al.  A nonlocal convection–diffusion equation , 2007 .

[11]  E. Chasseigne.  The Dirichlet problem for some nonlocal diffusion equations , 2007, Differential and Integral Equations.

[12]  Noemi Wolanski,et al.  How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems , 2006, math/0607058.

[13]  J. Coville,et al.  A nonlocal inhomogeneous dispersal process , 2007 .

[14]  Peter W. Bates,et al.  An Integrodifferential Model for Phase Transitions: Stationary Solutions in Higher Space Dimensions , 1999 .

[15]  Jérôme Coville,et al.  On a non-local equation arising in population dynamics , 2007, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[16]  P. Bates,et al.  Traveling Waves in a Convolution Model for Phase Transitions , 1997 .

[17]  Emmanuel Chasseigne,et al.  Asymptotic behavior for nonlocal diffusion equations , 2006 .

[18]  Rolf Rannacher,et al.  Trends in Nonlinear Analysis , 2002 .

[19]  Bashir Ahmad,et al.  Generalized quasilinearization method for a second order three point boundary-value problem with nonlinear boundary conditions , 2002 .

[20]  Adam J. J. Chmaj,et al.  The nonlocal bistable equation: Stationary solutions on a bounded interval ¤ , 2002 .

[21]  Jérôme Coville,et al.  Existence and Uniqueness of Solutions to a Nonlocal Equation with Monostable Nonlinearity , 2008, SIAM J. Math. Anal..

[22]  Xiaofeng Ren,et al.  HOMOCLINIC SOLUTIONS OF AN INTEGRAL EQUATION : EXISTENCE AND STABILITY , 1999 .

[23]  J. Coville.  Maximum principles, sliding techniques and applications to nonlocal equations , 2007 .

[24]  Julio D. Rossi,et al.  A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary , 2005 .

[25]  Jack Carr,et al.  Uniqueness of travelling waves for nonlocal monostable equations , 2004 .

[26]  V. Hutson,et al.  The evolution of dispersal , 2003, Journal of mathematical biology.

[27]  J. M. Mazón,et al.  A nonlocal p-Laplacian evolution equation with Neumann boundary conditions , 2008 .

[28]  On the perturbation of eigenvalues for the p-Laplacian , 2001 .

[29]  J. Rossi,et al.  Refined asymptotic expansions for nonlocal diffusion equations , 2008 .

[30]  Linghai Zhang,et al.  Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neuronal networks , 2004 .

[31]  Noemi Wolanski,et al.  Boundary fluxes for nonlocal diffusion , 2007 .

[32]  Fuensanta Andreu,et al.  The Neumann problem for nonlocal nonlinear diffusion equations , 2008 .

[33]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[34]  Gianna Stefani.  On Maximum Principles , 1991 .

[35]  Paul C. Fife,et al.  Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions , 2003 .

[36]  Xinfu Chen,et al.  Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations , 1997, Advances in Differential Equations.

[37]  Julián López-Gómez,et al.  First Variations of Principal Eigenvalues with Respect to the Domain and Point-Wise Growth of Positive Solutions for Problems Where Bifurcation from Infinity Occurs , 1998 .