I am motivated by the following question:

Given a uniform heat source in a convex domain, and suppose that the outside temperature is equal to $0$, can we determine where the long-time temperature inside the domain changes fastest on the boundary?

Mathematically, we let $\Omega \subset \mathbb{R}^n$ be a smooth convex domain, and let $u$ be the function satisfying \begin{align} \begin{cases} -\Delta u=1,\quad &\mbox{in $\Omega$}\\ u=0\quad &\mbox{on $\partial \Omega$}. \end{cases} \end{align} The question is, how to determine the set $$\Big\{p \in \partial \Omega: \Big|\frac{\partial u}{\partial \nu}(p)\Big|=\max_{\sigma \in \partial \Omega}\Big|\frac{\partial u}{\partial \nu}(\sigma)\Big| \Big\}?$$ My intuition is that, the set should coincide with the points on the boundary with minimal mean curvature. However, I cannot prove this claim. Any comments or ideas are really appreciated.

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