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Let $E$ be a regular hexagon centered at the origin of $\mathbb{R}^2$. Let $f$ be the harmonic function in $E$ with boundary value 1 on one of the sides of $E$ and boundary value $0$ on each of the remaining sides. What is the value of $f$ at the origin?

This question has shown up on an old PDE qual I am studying. This problem is causing me a lot of concern, because $f$ seems to be discontinuous on the boundary of $E$. But, given that $f$ is harmonic (and thus continuous) in $E$, shouldn't the boundary values of $f$ also define a continuous function?

Hints or explanations are greatly appreciated!

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I have undone the last edit. See Old John's answer below. –  Giuseppe Negro Jan 19 '13 at 21:19

2 Answers 2

up vote 7 down vote accepted

The solution is harmonic and thus analytic in the interior of $E$ and continuous up to the boundary, except at the end points of the segment where it is equal to 1.

When you rotate the segments, you'll get six functions of this form which are rotational images of each other (by uniqueness) and thus all have the same value at the origin. Adding these six functions, you get a harmonic function with boundary values equal to 1, hence it is 1 everywhere.

Therefore, $f(0) = 1/6$.

This is also the probability that a Brownian motion exits through one of the side of the hexagon assuming it starts at the origin (which could be turned into another proof).

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Edit: Please note that the OP has recently edited the question and removed the part that my comments below (to long to be a single comment) refer to.

Harmonic functions do have to be well-behaved inside a region where hey are defined, and in fact they have to be smooth there (infinitely differentiable), but the behaviour at the boundary can be much worse, and indeed as your question implies, it is quite possible to have a function which is harmonic at every point in the interior of the hexagon, but is discontinuous on the boundary itself.

For example, you might consider the harmonic measure. A good book on this is by Garnett: "Harmonic Measure" (Cambridge), and on page 1 he shows that the harmonic measure of an interval on the $x$-axis gives a harmonic function on the upper half-plane, but has boundary values which are $\pi$ inside the interval and zero elsewhere.

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I don't understand the relevance of this post to the question. I'm afraid this does not answer the question at all. –  Giuseppe Negro Jan 19 '13 at 21:12
    
Please take a look at the question as it was originally posted! The second half of his question (now deleted) was what I was commenting about, but it was too long to fit into a single comment. –  Old John Jan 19 '13 at 21:14
    
Of course, now I see. I have undone the last edit of the original question to prevent this misunderstanding to happen again. I wonder why the OP did that edit in the first place. –  Giuseppe Negro Jan 19 '13 at 21:18

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