Strong Localized Perturbations of Eigenvalue Problems

Abstract

This paper considers the effect of three types of perturbations of large magnitude but small extent on a class of linear eigenvalue problems for elliptic partial differential equations in bounded or unbounded domains. The perturbations are the addition of a function of small support and large magnitude to the differential operator, the removal of a small subdomain from the domain of a problem with the imposition of a boundary condition on the boundary of the resulting hole, and a large alteration of the boundary condition on a small region of the boundary of the domain. For each of these perturbations, the eigenvalues and eigenfunctions for the perturbed problem are constructed by the method of matched asymptotic expansions for small, where is a measure of the extent of the perturbation. In some special cases, the asymptotic results are shown to agree well with exact results. The asymptotic theory is then applied to determine the exit time distribution for a particle undergoing Brown|an motion inside a container having reflecting walls perforated by many small holes.

Stochastic Model of Acidification, Activation of Hemagglutinin and Escape of Influenza Viruses from an Endosome

“…For this purpose, we compare the reciprocal of the forward rate constant K (Equation 25), which is the mean time for a proton to bind a HA protein, with the free Brownian diffusion time scale. For a fixed proton concentration at a value c, the proton binding time is τ bind = 1 Kc , while the mean time for a proton to diffuse to the same binding site is [32][33][34][35] …”

Activation through a narrow opening

“…Matched asymptotics of two-and three-dimensional problems [22], [23], [24], [11] yield the leading term in the expansion of the principal eigenvalue in three dimensions and a full expansion in two dimensions. For the special case of the mixed Neumann problem with a small Dirichlet window in the boundary, the leading term obtained in [17], [18], [19] can be obtained by the application of the matched asymptotics expansion to this problem.…”

“…Here we consider the narrow escape problem for a Brownian motion in a field of force. The closely related problem of computing the principal eigenvalue of the Laplace operator for mixed boundary conditions on large and small pieces of the boundary was considered in [22], [23], [24], [11] (see section 6 for discussion).…”

Encounter dynamics of a small target by a polymer diffusing in a confined domain

“…Two different types of kinetics for diffusion-controlled processes in dense polymer systems have been discussed in 15 , depending on the root mean square displacement of the active polymer site (compact and non compact exploration). When the polymer is very small and its motion is dominated by its center of mass diffusion, then the narrow encounter time of the polymer (NETP) is precisely the mean first passage time for a Brownian particle to a small target, also known as the narrow escape time (NET [16][17][18][19][20][21][22][23][24] . In a space of dimension d, it is given by τ 2d = A πD ln 1 ε + O(1) for d = 2 (1)…”