I have had trouble when I try to prove the following:
For $u: \Omega \to \mathbb{R}$ be a harmonic function iff $u \in C^2(\Omega)$ and $\Delta u=\dfrac{\partial ^2u}{\partial x_1^2}+\ldots+\dfrac{\partial ^2u}{\partial x_n^2}\equiv 0 $
Any help will be appreciated.
You have defined a function to be harmonic if it satisfies the mean value property. For this problem, you will need the Poisson integral formula, and you will need to know that the function it produces satisfies the partial differential equation given in the problem statement. The following is a sketch of the proof that the mean value property implies your PDE.
Fix an arbitrary $z\in\Omega$. Find a closed ball $B$ centered at $z$ that lies in $\Omega$. Since the property we are interested in is local, it suffices to show that $u$ is harmonic in $B$. Using the Poisson kernel, we can construct a harmonic function $v$ in $B$ that agrees with $u$ on $\partial B$. Then $u-v$ is a harmonic function on $B$ that continuously extends to $\partial B$, where it is $0$. This means $u-v=0$ throughout $B$ using the maximum principle (which is proved using the mean-value property). So $u=v$, and $u$ satisfies the desired partial differential equation in the interior of $B$.
For more detail and the other direction, see Harmonic Function Theory by Axler, which you can download legally and for free at this link.