Consider the following theorem:
Assume $\varOmega\subset\mathbb{R}^{N} $ is an open region. Let $ u:\varOmega \to \mathbb{R} $ be a function. We say that $ u $ is harmonic in the weak way, if for any $ \delta(x) : \varOmega \to \mathbb{R} $ with compact support, twice continuously differentiable, the following holds
$$ \intop_{\varOmega}u\left(x\right)\varDelta\delta\left(x\right)dx=0 $$
We know that any weak harmonic function is also an harmonic function (in the regular way) in the same region.
Based on this theorem, given a region $ \varOmega=G\times\left(-\rho,\rho\right) $ where $\rho $ is a positive real number, and G is also an open region, and given a function $ u\left(x_{1},...,x_{N}\right):\overline{\varOmega}\to\mathbb{R} $ such that $ u $ is twice continuously differentiable and harmonic on $ \varOmega^{+}=G\times\left(0,\rho\right)$, and also for any $ x\in\varOmega $ we have the following: $ u\left(x_{1},...,x_{N-1},-x_{N}\right)=-u\left(x_{1},...,x_{N-1},x_{N}\right) $
How can I prove that $ u $ is harmonic on $ \varOmega $ ?