Existence of a local maximum on an open ball

Question

Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be a function of class $\mathcal{C}^2$. Let $D$ be an open ball in $\mathbb{R}^2$. We suppose there exists $m_0 \in D$ such that $\forall m \in \partial D, f(m_0) \geq f(m)$.

Show that there exists a local maximum $m_1$ of $f$ in $D$.

My proof so far:

1. I have shown that $cl(D)$ is a compact set.
2. Since $f$ is continuous and $cl(D)$ is a compact set, then $f:cl(D) \rightarrow \mathbb{R}$ reaches its supremum.

I guess what I have left to prove is that this supremum or upper bound is in D., but I don't know how.

Answer 1

So you already know that $f:cl(D)\to\mathbb{R}$ achieves global supremum. Recall that $\partial D$ and $D$ form a partitioning of $cl(D)$. Now $f(m_0)\geq f(m)$ for any $m\in\partial D$. And so if that global supremum is in $\partial D$ then $m_0\in D$ is a global supremum as well (by our inequality). Otherwise if the global supremum is not in $\partial D$ then it has to lie in the other piece, i.e. $D$.

And so we have that $f:cl(D)\to\mathbb{R}$ has a global supremum inside $D$. Now $D$ is open in $\mathbb{R}^2$ and thus the global supremum is a local supremum of the entire $f:\mathbb{R}^2\to\mathbb{R}$.

Answer 2

There exist $x$ in $\overline D$ such that $f(x) \geq f(y)$ for all $y \in \overline D$. If $x \in D$ we are done. If $x \in \partial D$ the $f(m_0) \geq f(x) \geq f(y)$ for all $y \in \overline D$ si $m_0$ so a local maximum.