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Some Properties of Laplacians on Fractals

✍ Scribed by Robert S. Strichartz

Publisher: Elsevier Science
Year: 1999
Tongue: English
Weight: 256 KB
Volume: 164
Category: Article
ISSN: 0022-1236
DOI: 10.1006/jfan.1999.3400

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✦ Synopsis

Kigami has defined an analog of the Laplacian on a class of self-similar fractals, including the familiar Sierpinski gasket. We study properties of this operator. We show that there is a maximal principle for solutions of certain nonlinear equations of the form 2u(x)=F(x, u(x)). We discuss the extension of the Laplacian to noncompact fractal blow-ups, and show that it is essentially self-adjoint, and we prove an analog of Liouville's theorem in some cases. We also give an explicit algorithm for solving the Dirichlet problem for certain domains in the Sierpinski gasket and give a characterization of all harmonic functions on those domains.

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