Analysis on Fractals

By Eric S. Mbakop. 2009

803 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/2ccc481cd035a4033ae8027f1415794569b11846

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces (2012, background): “Another case we are interested in arises if the regular Dirichlet form (E ,F) onX is induced by a regular resistance form (E ,F) on the set X , which is then equipped with the topology determined by the associated resistance metric, see [37, 38] and in particular [40, Definitions 3.”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “The existence of such a diffusion process Zt is wellknown when X is a Riemannian manifold (dw = 2; see [43]), the k-dimensional Sierpiński gasket (dw = log(k + 3)/ log 2 > 2; see [11]), more general nested fractals [41, 68], or the Sierpiński carpets [7, 8]; see [59] for more information.”
Fractal solids, product measures and fractional wave equations (2009, background): “…This journal is © 2009 The Royal Society2521 mathematicians began to look at partial differential equations—starting with Laplace’s or heat equation—on fractal (albeit non-random) sets (e.g. Kigami 2001; Strichartz 2006), an analogue of continuum physics and mechanics still needs to be developed.”
Geometrical Structure of Laplacian Eigenfunctions (2012, background): “A number of mathematical studies were devoted to the theory of partial differential equations on fractals in general and to localization of Laplacian eigenfunctions in particular (see [280, 477] and references therein).”
Heat Kernels and Green Functions on Metric Measure Spaces (2014, background): “There are a vast literature to devote to some of these questions, for example, in [2, 6, 13, 14, 15, 17, 32, 33, 34, 36, 37, 38, 39] for the Euclidean spaces or Riemannian manifolds, in [8, 10, 24, 25] for torus or infinite graphs, in [3, 5, 9, 27] for certain classes of fractals and in [12, 26, 28,”
Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates (2018, background): “7 There is a large class of fractal examples [59,60,70,92] with resistance forms E , a so-called Kusuoka measure μ, and a base of open sets O with finite boundaries, such that 1O ∈ BV (X) and ‖D1O‖ is absolutely continuous with respect to the counting measure on ∂O .”
Dynamical Systems on Spectral Metric Spaces (2010, background): “The earliest work recorded in the literature came in 1994, when Lapidus proposed to extend the Connes example of the Cantor set to fractals [43, 44].”
Uniqueness of Brownian motion on Sierpinski carpets (2008, unknown): “See [2, 23, 40] for further information and references.”