The sound of a drum with irregular boundary
The underlying question of this talk is "Can one hear the shape of a
drum?" which was posed by Mark Kac in 1966 (Am. Math. Mon.).
The sound which comes from the drum is linked to the eigenvalues of the
associated Dirichlet-Laplacian. Thus, the question amounts to studying
the spectrum of Laplace operators on domains with irregular (fractal)
boundaries. For such drums several conjectures have been made concerning
the above question. Notably the (modified) Weyl-Berry conjecture states
that from the sound one can recover the topological dimension of the
drum, its volume and the Minkowski dimension of its boundary as well as
the Minkowski content of its boundary.
We will examine the validity of the (modified) Weyl-Berry conjecture for drums with self-similar or self-conformal boundaries and discuss the geometric significance of the Minkowski dimension and content.
- Joint works with U.Freiberg and M.Kesseböhmer.