I am trying to solve this exercise :

Consider $U$ open , bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U)$ a harmonic function. Suppose that for each $x \in \partial U $ there exists a sequence $a_k$ in $U$ with

$$ a_k \rightarrow x$$ and

$$ \limsup \ u(a_k) \leq M$$

where $M$ is a constant independent of $x$. Show that $u \leq M $ in $U$.

To prove this exercise I am trying to use the maximum principle :

Consider $U$ an open, bounded set in $R^n$ with smooth boundary. Let $u \in C^2(U) \cap C(\overline{U})$ a harmonic function. Then the maximum and minimum occurs in the boundary of $U$. Someone can give me a hand to prove the exercise? Any help is appreciated.

Thanks in advance!

  • $\begingroup$ Do you mean $a_k\to x$? $\endgroup$ – user940 Sep 30 '13 at 2:17
  • $\begingroup$ yes. I fixed the error. thanks! $\endgroup$ – math student Sep 30 '13 at 2:21
  • $\begingroup$ In the case where $U$ is an interval $(0,1)$ you can get this result by applying the maximum principle to $(x_n,y_n)$ where $x_n \to 0,y_n \to 1$ are the sequences guaranteed by the hypothesis. I'm unsure how to generalise this at the moment but it's a starting point. $\endgroup$ – Anthony Carapetis Sep 30 '13 at 11:10
  • $\begingroup$ Where do you found this problem? $\endgroup$ – Tomás Sep 30 '13 at 14:46
  • $\begingroup$ @Tomás I found in a exam of my friend. Now me and my friends solved the exercise. Thanks for your attention =) $\endgroup$ – math student Oct 1 '13 at 1:10

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