Regularity of a domain - definition

Question

What does it mean when we say that a domain is $C^k$, $C^{k,\alpha}$, Lipschitz, or smooth? Is there an intuitive understanding?

Answer 1

Smooth domain $\Omega$ is an open and connected subset of the whole domain, say $\mathbb{R}^n$, of which the boundary $\partial \Omega$ is "smooth".

The smoothness of the boundary, intuitively speaking, is:

The boundary of a smooth domain can be viewed as the graph of a smooth function locally.

How smooth this function is determines the smoothness of the boundary. For example, $C^{k,\alpha}$-domain. Formal definition please refer to Gilbarg and Trudinger's book section 6.2. For any point on the boundary of a $C^{k,\alpha}$-domain, the boundary of domain within this point's neighborhood is the graph of a $C^{k,\alpha}$ function of $(n-1)$ variables. To visualize this a bit you could think the domain being in $\mathbb{R}^2$, a domain having $C^{k,\alpha}$ smoothness says each small piece of the boundary is a $C^{k,\alpha}$ function's graph, i.e., a smooth 1 dimension curve.

Another example would be Lipschitz domain, polygon and polyhedra are Lipschitz domains in $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively.

The motivation of formalization of the smooth domain originates from the idea of the estimates measuring how smooth the solution of a boundary value problem is.