Suppose that $D$ is a domain in $\mathbb{R}^n$ (that is, an open, bounded and connected subset), and that $u$ is an harmonic function on $D$. Let $x_0$ be a point at the boundary of $D$.
Question (vague version): Can we say something about the limit $\lim_{x \to x_0} u(x)?$
I guess that the short answer is no: that limit might exist as well as not exist, which happens, for example, if we plug a jumping initial datum into Poisson's integral formula on the ball. Clearly, what is giving us trouble is the possibility that $x$ approaches the boundary point $x_0$ in a tangential way. So here's a first refinement:
Question (refinement 1): Suppose that $D$ has a smooth boundary, so that it makes sense to speak of a normal vector field $\nu$ on $\partial D$. Is it true that the limit along normal lines, that is $\lim_{\varepsilon \to 0} u(x_0 - \varepsilon \nu(x_0))$ exists?
More generally:
Question (refinement 2): Can we develop a notion of nontangential limit of some kind so that $\text{non-t.-}\lim_{x\to x_0}u(x)=u(x_0)?$
Thank you.