SUMMING LOGARITHMIC EXPANSIONS FOR SINGULARLY PERTURBED EIGENVALUE PROBLEMS

Strong localized perturbations of linear and nonlinear eigenvalue prob lems in a bounded two-dimensional domain D are considered. The effects on an eigenvalue lambda0 of the Laplacian, and on the fold point lamb da(c0) of a nonlinear eigenvalue problem, of removing a small subdomai n D(epsilon), of ''radius'' epsilon, from D and imposing a condition o n the boundary of the resulting hole, are determined. Using the method of matched asymptotic expansions, it is shown that the expansions of the eigenvalues and fold points for these perturbed problems start wit h infinite series in powers of (-1/log [epsilond(kappa))]). Here d(kap pa) is a constant that depends on the shape of D(epsilon) and on the p recise form of the boundary condition on the hole. In each case, it is shown that the entire infinite series is contained in the solution of a single related problem that does not involve the size or shape of t he hole. This related problem is not stiff and can be solved numerical ly in a straightforward way. Thus a hybrid asymptotic-numerical method is obtained, which has the effect of summing these infinite logarithm ic expansions. Results obtained from the hybrid formulation are shown to be in close agreement with full numerical solutions to the perturbe d problems. The hybrid method is then used to determine the absorption time distribution for a particle performing Brownian motion in a doma in with reflecting walls containing several small absorbing obstacles.