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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.

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As a rule of thumb: that hypothesis allows you to prove theorems on simpler domains (e.g. semispaces, spheres) then transfer them via a change of coordinates over more general domains provided their boundary is sufficiently smooth. – Giuseppe Negro Jun 11 '11 at 17:39
I suspect in many cases the nature of the boundary isn't terribly important -- for example, think about the heat equation in the plane. The boundary does not need to be $C^2$ for the main theorems about heat flow. I think in many cases it just makes life a lot easier to assume the boundary is nice -- for the sake of proving theorems quickly without too much technical fussing-about. If you need more flexible boundary conditions, then you have to think more carefully about whatever PDE you're interested in. – Ryan Budney Jun 11 '11 at 17:40
up vote 3 down vote accepted

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 $x$.

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Why is there a "community wiki" tag after you answer? – Jack Jul 20 '11 at 15:48
What does the "uniform interior and exterior sphere conditions on the boundary" mean? I guess it's something like the uniform convergence of the inward and outward directional derivative defined on the boundary? – Jack Jul 20 '11 at 15:50
(a) I community wiki'd it because it is more of a summary + elaboration of what dissonance and Ryan Budney already mentioned in their comments. ... and I am somewhat surprised that the new SE FAQ no longer mentioned community wiki. The idea is that I don't get "credit" (in terms of rep and stuff) for this answer, and anybody is free to edit the answer (whereas for non-CW posts, there is a minimum rep requirement for editing other people's posts). – Willie Wong Jul 20 '11 at 16:24
(b) a point $x_0\in\partial\Omega$ is said to satisfy the interior sphere condition if there exists a point $x'\in\Omega$ and an open ball $B$ centered at $x'$ such that $B\subset \Omega$ and $x_0\in\partial B$. In other words, you can fit a (small) ball completely inside $\Omega$ such that $x_0$ sits on the boundary of the ball. The exterior sphere condition is analogous, but with the sphere sitting outside of $\Omega$. Uniformity means that one can give a lower bound of the radius of the relevant balls regardless of which point $x\in\partial\Omega$. – Willie Wong Jul 20 '11 at 16:29
+1. Now I see. Thanks for your elaboration. May I ask you to put the comment (b) to your answer? – Jack Jul 20 '11 at 21:36

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