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What is happening here : $\partial U $ is flat and $x^o \in \partial U $ is the center of the sphere, assume $\partial U $ to be flat near $x^0 $ and $x_n=0$ I am looking for a concrete example to appreciate whats happening when i define function the following way . Example can be in 1d or 2d . Thanks for ur help.

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$U$ is a open, bounded subset of $\mathbb R^n$. I think flat here means that the $\partial U$ divides the ball symmetrically . I am not pretty sure.

The text clip is from Partial Differential Equations by Lawrence C. Evans, p. 225, Chapter 5. In the second edition (p. 269, Section 5.4), $C^\infty$ has been replaced by $C^1$.

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    What is $U$? What does 'flat' mean?2012-07-16
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    @copper.hat : Sir , even i am not sure what flat means, but i have edited it as i have understood.2012-07-16
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    I don't know what you are trying to do. Perhaps if you gave a reference to the text you clipped?2012-07-16
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    @copper.hat : Sir , its from Evans PDE book . i have posted the reference .2012-07-16
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    I added a Google Books link to the second edition and a note about a change made in that edition.2012-07-16
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    @joriki : Thank you sir .2012-07-16

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It's a constructive way to extend a function $u$ through a flat part of the boundary to get a $C^1$ function $\bar u$. If to use reflection, defining $$\bar u(x_1,\ldots,x_{n-1},-x_n)=u(x_1,\ldots,x_{n-1},x_n),$$ then $\bar u$ for a continuous $u$ will be continuous in $B$. It's straightforward to check that $\bar u$ as defined in the book has all continuous derivatives of the first order in $B$. Actually only checking the continuity of $\partial u/\partial x_n$ when $x_n=0$ is needed. Analogously it is possible for $u\in C^m(\bar B_+)$ to obtain a function $$ \bar u(x_1,\ldots,x_{n-1},-x_n)=\sum_{k=1}^{m+1}\alpha_k u(x_1,\ldots,x_{n-1},x_n/k) $$ belonging the class $C^m(\bar B)$ with the proper choice of constants $\alpha_k$.

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The right kind of example to work through is $x_n^k$ for k=0,1,2... Of course the other variables don't matter here, so $x_n$ can be simply $x$. Plugging this into the reflection formula, you see that for k=0,1 it gives the same monomial. This indicates that the extension is $C^1$ smooth. For k=2 it no longer works; you'd need another term in the reflection formula to achieve $C^2$, as @Andrew wrote.