How does the boundary property usually work in PDE? This may be related to the question Is the hypersurface of class $C^k$ a $C^k$-differentiable manifold?. In almost every chapter of a PDE textbook(e.g. Folland's Introduction to Partial Differential Equations), the boundary of the domain is assumed to have some properties. The most common assumption is about the "smoothness"(I call it so, whether appropriate or not). For example, in Chapter 3 in Folland's book, there is a sentence as following:
"In this chapter $\Omega$ will be a fixed bounded domain in ${\mathbb R}^n$ with $C^2$ boundary $S$,..."
It seems that this assumption is always used implicitly. I can never understand how it actually work. I can only know superficially that $C^2$ means the coordinates charts are $C^2$-compatible in the definition of differentiable structure. Nothing more. Is there a rule of thumb that how this assumption is used?
It may be too vague to ask such question without a concrete proposition/theorem which has such assumption. Any suggestion for a good modification of the question will be really welcomed.
 A: One of the cases you need it is for prescribing boundary values. Just consider the change of variable formula. To define a $k$th order derivative space on the boundary of a set, you need that boundary to be described by a function that is at least $W^{k,\infty}$ (which is the Lipschitz space $C^{k-1,1}$). 
Similarly, the usual technique in proving boundary estimates for PDEs is to "straighten out" the boundary so that you can work in the case where your domain is the half-space. To do this change of variable, and preserve strong solutions in a uniform way, requires certain minimum differentiability assumptions on the boundary. 
Another case where it is used is that for a bounded domain with $C^2$ boundary, the boundary is compact, and therefore as a lower bound on the radius of curvature. This is then sufficient to guarantee uniform interior and exterior sphere conditions† on the boundary, and thus allows you to prove strong maximum principle for second order elliptic operators on the domain. 
† The sphere conditions are conditions on the "one sided" smoothness of the boundary. Let $\Omega$ be an open domain and let $x_0$ be a point on the boundary $\partial\Omega$. The boundary $\partial\Omega$ is said to satisfy an interior sphere condition with radius $r$ at $x_0$ if there exists $x'\in\Omega$ such that the open ball of radius $r$ around $x'$, which we denote $B_r(x')$, satisfies $B_r(x') \subset \Omega$ and $x_0 \in \partial B_r(x')$. Immediately you see that if $\partial\Omega$ satisfies the interior sphere condition with radius $r$, then it will also for all radii $r' < r$. The exterior sphere condition is defined analogously, with $x'$ and $B_r(x')$ required to reside to the exterior of $\Omega$. 
Observe that the sphere conditions give "one-sided" bounds on the "radius of curvature". For example, let $\Omega = \{ (x,y)\in \mathbb{R}^2, y > |x| \}$. Then at the corner at the origin, $\Omega$ does not satisfy the interior sphere condition for any $r$, but it satisfies exterior sphere conditions for any positive radius. 
Lastly, uniformity is just the statement that $\exists r_0$ st. $\forall x\in\partial\Omega$, $\partial\Omega$ satisfies an interior/exterior sphere condition of radius $r_0$ at $x$. 
