Harmonic functions in $\mathbb{R}^d$

Question

I want to establish the equivalence of the 3 standard definitions, and that harmonic functions are $C^\infty$. The 3 definitions are:

1. Mean value property and continuous.
2. $C^2$ and $0$ Laplacian.
3. 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.

Answer 1

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:

$1.u\in C^\infty (B^\circ (0,r))$

$2.\Delta u=0,\text{ in } B^\circ (0,r)$

$3.\lim\limits_{x\to x_0}u(x)=g(x_0),\forall x_0\in\partial B(0,r)$