Scaling of curvature measures and the modified Weyl-Berry conjecture
Final Report Abstract
The main subject of the project were geometric measures and functionals for the quantitative description of deterministic and random fractals. The aims were 1. to investigate new variants of these geometric functionals based on approximation with unions of boxes instead of parallel sets; 2. to extend the class of fractal models for which such geometric functionals are usable, and 3. to investigate the relevance of these geometric measures for the spectral asymptotics of the Laplace operator (i.e., for sound and heat conduction) on domains with fractal boundary. Sustainable successes were achieved in all three parts of the project. The functionals investigated are based on the approximation of the fractal sets to be described with parallel sets or with unions of boxes. They are defined as rescaled limits of measures known from classical geometry. These include volume and surface measures, higher-order curvature measures and support measures. Existence results for such functionals (as well as formulas for their calculation) could be extended to further classes of fractals, including to so-called V-variable random fractals and generalisations thereof. It turned out that only mean values can be considered there, but the associated fractal parameters are structurally similar to those for classical self-similar sets. This will facilitate their application, but numerical methods have yet to be developed. An important step towards spectral asymptotics was taken by determining the associated fractal curvatures for regions with piecewise self-similar boundaries. In the classical smooth case as well as for very special fractal domains, it is known from the literature that these functionals are related to the asymptotics in heat conduction problems. This is now being further investigated for the abovementioned domains. Further progress has been made with refined tube formulae for fractals in the sense of the theory of complex dimensions, in which the coefficients can now be described by support measures.
Publications
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The Mean Minkowski Content of Homogeneous Random Fractals. Mathematics, 8(6), 883.
Zähle, Martina
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Almost sure convergence and second moments of geometric functionals of fractal percolation. Advances in Applied Probability, 56(3), 927-959.
Klatt, Michael A. & Winter, Steffen
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Lectures on Fractal Geometry. Fractals and Dynamics in Mathematics, Science, and the Arts: Theory and Applications. WORLD SCIENTIFIC.
Zähle, Martina
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Mean Lipschitz–Killing curvatures for homogeneous random fractals. Journal of Fractal Geometry, Mathematics of Fractals and Related Topics, 10(1), 1-42.
Rataj, Jan; Winter, Steffen & Zähle, Martina
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Mean Minkowski and $s$-contents of $V$-variable random fractals. Asian Journal of Mathematics, 27(6), 955-970.
Zähle, Martina
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On volume and surface area of parallel sets. II. Surface measures and (non)differentiability of the volume. Bulletin of the London Mathematical Society, 57(3), 895-912.
Rataj, Jan & Winter, Steffen
