Summary: We review some recent work of the author (and his collaborators) regarding harmonic analysis and partial differential equations on fractals; more specifically, the vibrations of “fractal drums” and the spectral distribution of Laplacians on (suitable) self-similar fractals. We also discuss how this work was combined by the author with techniques from Connes’ noncommutative geometry to construct “volume measures” on such fractals, including an analogue in this context of the Riemannian volume measure. Further, we announce new results regarding the nature of these volume measures and, in special cases, their relationship with a suitable notion of Hausdorff measure. In addition, we consider the notion of “complex dimension” of (self-similar) fractals and its relationship with oscillatory phenomena in our setting. We also propose and discuss a number of conjectures and open problems in this area, aimed at further developing geometric analysis on fractals and – in the long term – at laying out some of the foundations for what may be coined “noncommutative fractal geometry”.
Although we stress mostly the mathematical aspects in this paper, we mention that this work – along with its proposed extensions – has natural physical motivations and applications, including transport in porous media, percolation networks, the vibrations of fractal resonators (such as certain microwave cavities), as well as diffusion and wave propagation in fractally structured media.
For the entire collection see [Zbl 0872.00037].