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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.

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Do you want these limits to exist at every point or at almost every point? And do you want to assume any kind of regularity, boundedness, or something? –  Lukas Geyer Dec 1 '12 at 0:11
    
Refinement 1 is stringent: smooth boundary, limit at all points. But what I want to say with refinement 2 is that I'm interested also in results of a different kind, such as limits at almost all points of the boundary and the like. –  Giuseppe Negro Dec 1 '12 at 0:13

1 Answer 1

up vote 2 down vote accepted

You can always build counterexamples to the existence of the radial limit at all points, even in very nice domains and for bounded harmonic functions. Here is one construction using complex analysis.

Let $D$ be the unit disk in the complex plane, and let $G = D \setminus S$, where $S$ is the spiral given in polar coordinates by $r = 1-e^{-\theta}$, where $0 \le \theta < \infty$. (This is a spiral contained in $D$ which accumulates on the whole boundary $\partial D$.) By the Riemann Mapping Theorem there exists a conformal map $f:D \to G$, and by some standard facts about boundary behavior of conformal maps there exists $z_0 \in \partial D$ such that the image of the radius $\{f(rz_0): 0\le r<1$} accumulates on the whole boundary $\partial D$. Now this implies that $u(z) = \operatorname{Re} f(z)$ is a bounded harmonic function in the unit disk $D$ for which the image of the radius $\{u(rz_0)\}$ accumulates on the whole interval $[-1,1]$, so that in particular the radial limit at $z_0$ does not exist.

There are positive results about the existence of these limits almost everywhere for bounded harmonic functions, or for Hardy spaces. These start from Fatou's theorem in the unit disk, but they can be pretty easily generalized to domains with nice boundaries.

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I see, thank you very much. I had this intuition that the radial limit was forced to exist but your answer makes it clear that the situation is more delicate than that. –  Giuseppe Negro Dec 1 '12 at 0:43

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