Differential Equations on Fractals opens the door to understanding
the recently developed area of analysis on fractals, focusing on the
construction of a Laplacian on the Sierpinski gasket and related
fractals. Written in a lively and informal style, with lots of
intriguing exercises on all levels of difficulty, the book is accessible
to advanced undergraduates, graduate students, and mathematicians who
seek an understanding of analysis on fractals. Robert Strichartz takes
the reader to the frontiers of research, starting with carefully
motivated examples and constructions.
One of the great accomplishments of geometric analysis in the
nineteenth and twentieth centuries was the development of the theory of
Laplacians on smooth manifolds. But what happens when the underlying
space is rough? Fractals provide models of rough spaces that
nevertheless have a strong structure, specifically self-similarity.
Exploiting this structure, researchers in probability theory in the
1980s were able to prove the existence of Brownian motion, and therefore
of a Laplacian, on certain fractals. An explicit analytic construction
was provided in 1989 by Jun Kigami. Differential Equations on Fractals
explains Kigami's construction, shows why it is natural and important,
and unfolds many of the interesting consequences that have recently been
discovered.
