Abstract
It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V (0), which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V (0). In this paper, I prove that, provided an eigenform exists, even if the form on V (0) is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of Γ-convergence (but these two limits can be different). The problem of Γ-convergence was first studied by S. Kozlov on the Gasket.
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Peirone, R. Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals. Potential Analysis 21, 289–309 (2004). https://doi.org/10.1023/B:POTA.0000033332.12622.9d
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DOI: https://doi.org/10.1023/B:POTA.0000033332.12622.9d