Induced Dirichlet forms on Self-similar sets

Induced Dirichlet forms on Self-similar sets

Abstract

Fractals are some irregular and complex objects tl1at often possess self—similarity, infinite complexity, and fractional dimension. They arise everywhere in nature, bio-logical science a11d human society: coastlines, seismic data, distributions of galaxies, microscopic surface of materials, blood vessels and lungs, DNA patterns and stock market charts. Fractals are used by scientists and engineers to compress images and generate realistic computer graphics, to build cell—pl1one antennas, and to locate oil or geologic faults. Computing fractal dimension can help detect malignant cells in the medical sector. Fractals also arise naturally in many branches of mathematics, such as dynamical systems, analysis, partial differential equations, geometry and number theory.

In this proposal, we study Dirichlet forms (equivalently, energy forms) on fractals. The theory includes the important Laplace operator. As is known, in classical anal-ysis, the theories of heat diffusion, wave propagation and all considerations in partial differential equations depend on the Laplacian. The operator is differential i11 nature while fractals are highly non—smooth, a11d hence many classical perceptions need to be adjusted. Therefore defining the Laplacian on fractals and studying its properties are some central problems in the subject. The projects in the proposal include the search for ways to construct and study the Dirichlet form and Laplace operator on self—similar sets by making use of so111e probabilistic and analytic techniques.