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Consider $B_1 = B(0,1)$ the unitary ball of $R^n.$ Denote $B^{+} = \{ x \in B_1 ; x_n > 0\}$ and $B_{ - } = \{ x \in B_1 ; x_n \leq 0\}$. Let $u \in L^{\infty}_{loc}(B^{+}) \cap W^{2,2}(B^{+}) \cap L^{1}(B^{+}) $ with $\Delta u = f $ in the weak sense for some $f \in L^{2}(B^{+}) $. Suppose that $u \phi \in W^{1,2}_{0}(B^{+}) $ for any cutoff function$\phi \in C^{\infty}_{0}(B^{+}) $

Extend $u$ to $B_{-} $ by odd reflection, and consider $\overline{u}$ this extension. This new function is in $W^{2,2}(B_1)$ with $\Delta \overline{u} = \overline{f}$ for some $\overline{f} \in L^{2}(B_1)$?

Intuitively this is true, but i have no idea of a proof. Someone could give me a help or point a reference?

thanks in advance!

math student
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1 Answers1

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Nothing in your assumptions prevents $u$ from being nonzero on the plane of reflection. That would create a jump discontinuity in the odd reflection. This problem can be avoided by using a higher order reflection.

Generally, to show that some process of reflection preserves some $W^{k,p}$ space, one considers the effect of reflection on smooth functions. If the reflected function is in $W^{k,p}$, with the Sobolev norm controlled by the norm of original function, then the reflection operator is bounded from the space $W^{k,p}$ to itself. In practice, the part about "Sobolev norm controlled" is trivial, and the only real issue is whether the derivatives of orders up to $k-1$ are continuous across the plane of reflection.

For example, the aforementioned higher order reflection $f(-x)=4f(x/2)-3f(x)$ maps $C^1$ functions into $C^1$ functions, and the integral of $|f''(-x)|^p$ is bounded by a multiple of the integral of $|f''(x)|^p$. Therefore, this reflection preserves $W^{2,2}$. (I used notation for one-dimensional case here, but the process is the same.)

Returning to odd reflection: If the function to be reflected is in $W^{2,2}_0$ of the upper half, then reflection preserves $W^{2,2}$. One argues as above, but only needs to consider smooth functions vanishing on the boundary, and those stay continuous under odd reflection.

  • now I noted that i forget to write $u \phi \in W^{1,2}{0}(B{+})$ for all cut off function $\phi \in C^{\infty}{0}(B{+})$.pardon me for this . with this assumption the affirmation will be true? – math student Oct 07 '14 at 22:28
  • I don't see how vanishing of $u\phi$ helps, if you are reflecting $u$ itself. Just imagine $u$ that is equal to $1$ on some part of the reflection plane; this does not contradict $u\phi\in W_0^{1,2}(B+)$. –  Oct 07 '14 at 22:34
  • you re right . thanks! – math student Oct 07 '14 at 22:35