Analysis, Probability and Mathematical Physics on Fractals

edited by Patricia Alonso Ruiz, Joe P Chen, Luke G Rogers, Robert S Strichartz and Alexander Teplyaev

In the 50 years since Mandelbrot identified the fractality of coastlines, mathematicians and physicists have developed a rich and beautiful theory describing the interplay between analytic, geometric and probabilistic aspects of the mathematics of fractals. Continue reading

Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

by Patricia Alonso-Ruiz, Fabrice Baudoin, Li Chen, Luke Rogers, Nageswari Shanmugalingam, Alexander Teplyaev

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. Continue reading

The Strong Maximum Principle for Schrödinger operators on fractals

by Marius V. Ionescu, Kasso A. Okoudjou, Luke G. Rogers

We prove a strong maximum principle for Schrödinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.

Arxiv preprint version

Published version in Demonstratio Mathematica

Power dissipation in fractal AC circuits

by Joe P Chen, Luke G. Rogers, Loren Anderson, Ulysses Andrews, Antoni Brzoska, Aubrey Coffey, Hannah Davis, Lee Fisher, Madeline Hansalik, Stephen Loew and Alexander Teplyaev.

We extend Feynman’s analysis of an infinite ladder circuit to fractal circuits, providing examples in which fractal circuits constructed with purely imaginary impedances can have characteristic impedances with positive real part. Continue reading

Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals.

by Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev

We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth
functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat
operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. Continue reading

Generalized Eigenfunctions and a Borel Theorem on the Sierpinski Gasket

by Kasso A Okoudjou, Luke G. Rogers and Robert S. Strichartz

We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet. Continue reading