AbstractsMathematics

We study analysis and partial differential equations on metric measure spaces by investigating the property of Sobolev functions or Sobolev mappings and studying the viscosity solutions to some partial differential equations. This manuscript consists of two parts. The first part is focused on the theory of Sobolev spaces on metric measure spaces. We investigate the continuity of Sobolev functions in the critical case in some general metric spaces including compact connected one-dimensional spaces and fractals. We also constructe a large class of pathological n-dimensional spheres in ℝn+1 by showing that for any Cantor set Csubsetℝn+1 there is a topological embedding f: