Sobolev Spaces and Elliptic Equations

Lipschitz domains. Our presentations here will almost exclusively be for bounded Lipschitz domains. Roughly speaking, a domain (a connected open set) Ω ⊂ R is called a Lipschitz domain if its boundary ∂Ω can be locally represented by Lipschitz continuous function; namely for any x ∈ ∂Ω, there exists a neighborhood of x, G ⊂ R, such that G ∩ ∂Ω is the graph of a Lipschitz continuous function under a proper local coordinate system. Of course, all smooth domains are Lipschitz. A significant non-smooth example is that every polygonal domain in R or polyhedron in R is Lipschitz. A more interesting example is that every convex domain in R is Lipschitz. A simple example of non-Lipschitz domains is two polygons touching at one vertex only.