[Math] Reflection from the upper half ball to the whole ball is harmonic

harmonic functionspartial differential equations

I have a question about problem 9(b) in Chapter 2 of Evans' PDE book. It says if we have $u$ is harmonic in the open upper half ball $U^+$ and $u\in C^2(U^+)\cap C(\bar{U^+})$, $u=0$ for $x\in \bar{U^+}\cap\{x_n=0\}$ Then by the reflection principle we define$$v(x)=u(x)\phantom{fd} \text{ if }\phantom{f}x_n\geq0$$$$v(x)=-u(x_1,x_2…-x_n)\phantom{fd} \text{ if }\phantom{f}x_n<0$$

Then why is $v$ in $C^2(U^+)$? I guess once we show this, we can show $v$ is harmonic. The hint is to apply Poisson's formula for the ball. But I still don't know how to do it. I am not sure how to deal with those points on the axis.

Best Answer

Indeed, it is not obvious that the extended function is $C^2$. But you don't have to prove harmonicity by checking the definition at the beginning of the chapter: there are other results you could use.

For example, there is a characterization of harmonicity that does not involve $C^2$: Theorem 6 (1st edition) states that a continuous function that has the mean value property (for all spheres contained in the domain) is harmonic. Unfortunately, it's difficult to apply this result here, because of the spheres that intersect the hyperplane of reflection.

So, let's use the hint instead. Let $w$ be the Poisson integral involving the values of $v$ on a slightly smaller concentric ball $W\subset U$. Then

  1. $w$ is harmonic in $W$ — there is a theorem in the chapter about that.
  2. $w(x_1,\dots,x_{n-1},-x_n) = -w(x_1,\dots,x_{n-1},-x_n)$, because the boundary values have this property (the sum of $w$ and its reflection is a harmonic function vanishing on the boundary).
  3. In particular, $w=0$ on the hyperplane.
  4. In each open half-ball $W^+$ and $W^-$, the equality $w=v$ holds (apply the maximum principle to $w-v$)
  5. Since $w=v$ everywhere in $W$, it follows that $v$ is harmonic in $W$.
  6. Since the radius of $W$ could be arbitrarily close to the radius of $U$, it follows that $v$ is harmonic in $U$.
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