Shine two lazers on a vibrating rectangular surface. An interference pattern appears. Each dark or light line is a level set of a mode shape. We solve the problem: Use the level set pattern to find properties of the object. Asymptotic analysis produces simple formulas that provide estimates for the unknown properties.

We briefly describe the analysis and give a sample of the results.

We study an initial-boundary value problem for a fully nonlinear system which describes the propagation of plane shear waves in a three-dimensional soil, assuming a hypoplastic flow rule. This model was derived to study liquefaction of soils. For a periodic square-wave velocity disturbance on the boundary, the solution is found to saturate away from the boundary. We calculate the asymptotic state, in terms of the initial and boundary data. This formula has a fractal behavior, as the shape of the square wave is varied.

We survey the history of semipositone problems and discuss recent progress for systems. Many open problems will also be discussed.

Existence and stability of spatially nonconstant solutions is one of the most interesting topics in the theory of reaction-diffusion equations. In 1975, it was shown by N. Chafee that any nonconstant stationary solution of

is unstable. Since then, many related results have been obtained for higher dimensional domains, spatially inhomogeneous and/or time-periodic equations, more general order-preserving systems, and so on. In this talk, I will describe some recent results for this problem, especially in the case of non-simply-connected domains.

Last modified October 12, 1998.
Copyright © A. J. Meir, all rights reserved.