Potential and scattering theory on wildly perturbed domains

Abstract

We study the potential, scattering, and spectral theory associated with boundary value problems for the Laplacian on domains which are perturbed in very irregular fashions. Of particular interest are problems in which a "thin" set is deleted and the behavior of the Laplace operator changes very little, and problems where many tiny domains are deleted. In the latter case the "clouds" of tiny obstacles may tend to disappear, to solidify, or to produce an intermediate effect, depending on the relative numbers and sizes of the tiny domains. These phenomena vary according to the specific boundary value problem and in many cases their behavior is contrary to crude intuitive guesses.