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arXiv · 1411.2491

A note on measure-geometric Laplacians

Marc Kesseböhmer·Tony Samuel·Hendrik Weyer
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Abstract

We consider the measure-geometric Laplacians $\Delta^{\mu}$ with respect to atomless compactly supported Borel probability measures $\mu$ as introduced by Freiberg and Z\"ahle in 2002 and show that the harmonic calculus of $\Delta^{\mu}$ can be deduced from the classical (weak) Laplacian. We explicitly calculate the eigenvalues and eigenfunctions of $\Delta^{\mu}$. Further, it is shown that there exists a measure-geometric Laplacian whose eigenfunctions are the Chebyshev polynomials and illustrate our results through specific examples of fractal measures, namely Salem and inhomogeneous self-similar Cantor measures.

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Marc Kesseböhmer, Tony Samuel, Hendrik Weyer. 2014-11-10. A note on measure-geometric Laplacians. https://doi.org/10.1007/s00605-016-0906-0

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