# a version of the comparison lemma (harmonic functions)

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.

• Do you mean $a_k\to x$? – user940 Sep 30 '13 at 2:17
• 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. – Anthony Carapetis Sep 30 '13 at 11:10