While studying about Green's identities from PDE by L.C.Evans I came across the definition of a $C^k$ boundary ( $dU$ ) of a bounded and open set ( $U$ ) in euclidean space . Then comes the following definitions :
1) If $dU$ is $C^1$ , then along $dU$ is defined the outward pointing unit normal vector field $v$ .
2) Let $u \in C^1 (\overline U) $ .We call
$$ {\partial U \over \partial v} =v.Du$$
the outward normal derivative of $u$.
My questions are:
1) In the first definition , what is the need for $dU$ to be $C^1$ to be able to define the outward pointing normal vector? I'm not able to relate the definition of a $C^1$ boundary to the fact that we can define an outward pointing normal vector field.
2) Why should $u$ be continuous upto the boundary, when all we need for the definition is the existence of all partial derivatives ?
Best Answer
See this question for an explanation of how $C^1$ smoothness is used to define the normal vector field. As littleO said, there is not a natural concept of "normal" at a corner of a square, for example.
For question 2): In a smooth domain, the continuity of derivatives up to the boundary implies the continuity of the function up to the boundary (via a form of mean value theorem). So we don't lose anything by making the definition what it is. Anyway, it is not usually the goal of a mathematician to make definitions as general as possible; they should be as useful as possible. Usefulness and generality are not the same thing, especially when writing a textbook.