If $\Delta u \geq 0$, then $u$ attains its maximum in the boundary $\partial \Omega$

Asked May 17 '17 at 22:31

Active May 17 '17 at 22:39

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Let $\Omega \subset \mathbb R^n$ a bounded set and let $u \in C^2(\Omega) \cap C(\overline\Omega)$ be a real-valued function.

I need a hint to proof that if the laplacian $\Delta u \geq 0$, then $u$ attains its maximum in the boundary $\partial \Omega$.

Help?

edited May 17 '17 at 22:39

asked May 17 '17 at 22:31

user 242964

2

This is answered in Sign of Laplacian at critical points of $\mathbb R^n$. If you just want a hint: if $a$ is an interior maximum point of $\Delta u$ then try to use the second derivative test for functions of one variable one variable at the time to show that $\Delta u \leq 0$ in this case. – Winther May 17 '17 at 22:49

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