TITLE: Harmonic functions in $\mathbb{R}^d$
QUESTION [5 upvotes]: I want to establish the equivalence of the 3 standard definitions, and that harmonic functions are $C^\infty$.  The 3 definitions are:

Mean value property and continuous.
$C^2$ and $0$ Laplacian.
Mean value property on arbitrarily small balls and continuous.

The only help I think I need here is proof of the existence and uniqueness of solution for the Dirichlet problem on the ball. (That would help me to get 3 implies 2, along with the maximal property for harmonic functions of type 3, which should be easy to prove.)  2 implies 1 by using arguments to take a derivative under the integral, and then using gauss' divergence theorem.  1 implies 3 obviously.  I also need help seeing that harmonic functions must be $C^\infty$.  I am not used to methods that don't involve complex analysis, as must be used here.
I know that there is a theory of plurisubharmonic functions.  As a bonus question, do those tend to be useful outside of complex analysis, and are they ever discussed for odd dimension?  For example, I have never seen them discussed in harmonic analysis, nor do they seem to be useful in relation to Brownian motions, which is why I am learning the d-dimensional version of harmonic function theory now.
Edit: Come to think of it, I'd also like some help proving the open mapping property and maximal properties for harmonic functions.  Please only assume definition 3 here, because I will use it and a connectedness argument to establish 3 implies 2.  The precise statement of definition 3 is as in Greene and Krantz but for d dimensions:
$f$ is "harmonic (3)" if $f$ is continuous and $\forall x \in U$ the domain of $f$ there exists $\epsilon>0$ such that all balls of radius $\epsilon$ or less centered at $x$ are contained in U, and $f$ satisfies the MVP for that ball/spherical shell.

REPLY [2 votes]: For the Dirichlet problem on a Ball in $\mathbb R^n$ of radius $r$: 
$$\begin{cases}\Delta u=0&\text{ in }B^\circ (0,r)\\u=g&\text{ in }\partial B(0,r)\end{cases}$$ assuming $g$ is continuous on $\partial B(0,r)$. 
The Poisson Kernel on this Ball is $$K(x,y)=\dfrac {1}{na(n)r}\cdot\dfrac {r^2-|x|^2}{|x-y|^n}$$ where $a(n)$ is the volume of the unitary ball. 
Let $$u(x)=\dfrac{r^2-|x|^2}{na(n)r}\int_{\partial B(0,r)}\dfrac{g(y)}{|x-y|^n}dy, x\in B^\circ (0,r)$$
Then you have to prove that: 
$1.u\in C^\infty (B^\circ (0,r))$ 
$2.\Delta u=0,\text{ in } B^\circ (0,r)$
$3.\lim\limits_{x\to x_0}u(x)=g(x_0),\forall x_0\in\partial B(0,r)$