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 Gamma-convergence (but these two limits can be different). The problem of Gamma-convergence was first studied by S. Kozlov on the Gasket.

Peirone, R. (2004). Convergence of discrete Dirichlet forms to continuous Dirichlet forms on fractals, 21(3), 289-309 [10.1023/B:POTA.0000033332.12622.9d].

Convergence of discrete Dirichlet forms to continuous Dirichlet forms on fractals

PEIRONE, ROBERTO
2004-01-01

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 Gamma-convergence (but these two limits can be different). The problem of Gamma-convergence was first studied by S. Kozlov on the Gasket.
2004
Pubblicato
Rilevanza internazionale
Articolo
Sì, ma tipo non specificato
Settore MAT/05 - ANALISI MATEMATICA
English
fractals; Dirichlet forms; Gamma and pointwise convergence
Peirone, R. (2004). Convergence of discrete Dirichlet forms to continuous Dirichlet forms on fractals, 21(3), 289-309 [10.1023/B:POTA.0000033332.12622.9d].
Peirone, R
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/31307
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