Mathematical Sciences: Spectral and Fractal Geometry for Variational Elliptic Boundary Value Problems

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The relationship between the geometry of a manifold and the invariants of the elliptic differential operators on it is one of the main unifying themes of contemporary mathematical research. (For instance, the natural frequencies of a vibrating membrane constitute the spectrum of a differential operator called the Laplacian; knowing the spectrum, one can infer a great deal about the shape of the membrane.) The case of a region with fractal boundary, so irregular that its dimension is no longer an integer, has been the setting for some recent breakthroughs relating spectral information to geometry. In particular, thanks to work of Professor Lapidus and a French colleague of his, it is now known that one can recover the dimension of the boundary from the spectrum of the Laplacian in such a region. This work, which has involved both computer experimentation and sophisticated mathematical technique, has applications to the study of porous media and the scattering of waves from fractal surfaces. Its continuation is the main thrust of the project supported by this award. More precisely, Lapidus will investigate sharpened forms of the asymptotic formula for the eigenvalue distribution of Laplace operators with various boundary conditions. The goal is to make yet clearer the role of the Minkowski dimension of the boundary in this distribution. There are analogous formulas for the trace of the associated heat semigroup. The case in which not only the boundary of the domain, but the domain itself, is fractal will be studied.