Potential Theory on Sierpi?ski Carpets

This self-contained book lays the foundations for a systematic understanding of potential theoretic and uniformization problems on fractal Sierpi?ski carpets, and proposes a theory based on the latest developments in the field of analysis on metric spaces. The first part focuses on the development of an innovative theory of harmonic functions that is suitable for Sierpi?ski carpets but differs from the classical approach of potential theory in metric spaces. The second part describes how this theory is utilized to prove a uniformization result for Sierpi?ski carpets. This book is intended for researchers in the fields of potential theory, quasiconformal geometry, geometric group theory, complex dynamics, geometric function theory and PDEs.