Analysis on hybrid fractals

Ruiz, Patricia Alonso; Chen, Y.; Gu, Hang; Strichartz, Robert S.; Zhou, Zhengzheng

doi:10.3934/cpaa.2020004

Cited by 5 publications

References 26 publications

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Graph‐like spaces approximated by discrete graphs and applications

Post

Simmer

2021

Mathematische Nachrichten

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We define a distance between energy forms on a graph‐like metric measure space and on a suitable discrete weighted graph using the concept of quasi‐unitary equivalence. We apply this result to metric graphs, graph‐like manifolds (e.g. a small neighbourhood of an embedded metric graph) or pcf self‐similar fractals as metric measure spaces with energy forms associated with canonical Laplacians, e.g., the Kirchhoff Laplacian on a metric graph resp. the (Neumann) Laplacian on a manifold (with boundary), and express the distance of the associated energy forms in terms of geometric quantities. In particular, we show that there is a sequence of domains converging to a pcf self‐similar fractal such that the corresponding (Neumann) energy forms converge to the fractal energy form. As a consequence, the spectra and suitable functions of the associated Laplacians converge, the latter in operator norm.

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Graph‐like spaces approximated by discrete graphs and applications

Post

Simmer

2021

Mathematische Nachrichten

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Embedded trace operator for infinite metric trees

Franceschi,

Naderi,

Pankrashkin

2024

Mathematische Nachrichten

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We consider a class of infinite weighted metric trees obtained as perturbations of self‐similar regular trees. Possible definitions of the boundary traces of functions in the Sobolev space on such a structure are discussed by using identifications of the tree boundary with a surface. Our approach unifies some constructions proposed by Maury, Salort, and Vannier for discrete weighted dyadic trees (expansion in orthogonal bases of harmonic functions on the graph and using Haar‐type bases on the domain representing the boundary), and by Nicaise and Semin and Joly, Kachanovska, and Semin for fractal metric trees (approximation by finite sections and identification of the boundary with a interval): We show that both machineries give the same trace map, and for a range of parameters we establish the precise Sobolev regularity of the traces. In addition, we introduce new geometric ingredients by proposing an identification with arbitrary Riemannian manifolds. It is shown that any compact manifold admits a suitable multiscale decomposition and, therefore, can be identified with a metric tree boundary in the context of trace theorems.

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