Extension theorems in Sobolev spaces: Solving for constants

Question

I saw this problem in PDE book and tried searching for the idea behind solving it which I have not been able to find yet.

If we have $n\ge2$, $B=\{x\in\mathbb R^n:|x|<1\}$ and $B_+=B\cap\{x_n>0\}$. Also, for $u\in C_c^2 (\bar B_+)$, the extension $Eu(x)=u(x)$ if $x_n>0$ and $Eu(x)=\sum_{j=1}^3 \lambda_j u(x^\prime,{-j\over 3}x_n)$ if $x_n<0$ defines function $Eu\in C_c^2(B)$. How would you then solve for constants $\lambda_1, \lambda_2, \lambda_3$?

Any help or idea would be much appreciated.

Answer 1

To preserve the $C^k$ smoothness, the extension process must preserve polynomials of $x_n$ of degrees up to $k$. (Indeed, any change of coefficients of such a polynomial breaks the continuity of its derivatives.) This requires:

• $\sum_j \lambda_j=1$ to preserve $x_n^0$.
• $\sum_j (j/3)\lambda_j=1$ to preserve $x_n^1$.
• $\sum_j (j/3)^2\lambda_j=1$ to preserve $x_n^2$.

These three linear equations uniquely determine the constants $\lambda_j$, $j=1,2,3$. To verify that the extension indeed preserves the class $C^2$, it's probably easiest to differentiate $Eu$ and check that the derivative is continuous.

You can find examples of such computations on my blog: Higher-order reflections.