Suppose $D := \{(x, y) \in \Bbb{R}^2: x^2+y^2 \leq 1\}$ is the closed unit disk, and $\Gamma$ be the class of functions
$$ \Gamma := \{f: D \to \Bbb{R} \text{ such that } f \in C^1(D) \text{ and } |f| \leq 1 \} $$
that is continuously differential in the closed unit disk, i.e., that is continuously differential in some open set containing $D$, and with bounds $1$. Then what is the supremum of a minimum of the norm of the gradient of such a $f$?
$$ M := \sup_{f \in \Gamma}\min_{x \in D} \|\nabla f\|_2 $$
This question originates from a multivariable calculus problem, which proves that $M \leq 16$ by constructing a function $$g(x,y) = f(x,y)+ 2(x^2+y^2)$$ and from $g(\partial D) \geq 1, g(0,0) \leq 1$ one has the minimum of $g$ is in the interior $D^\circ$ of $D$. Hence some point $(x_0, y_0) \in D^\circ$ gives $\nabla g (x_0, y_0)= 0$ and $\nabla f (x_0, y_0) = -4(x_0, y_0)$ and $\|\nabla f (x_0, y_0)\| = 16(x_0^2+y_0^2) < 16$.
I suspect that $M$ cannot be $16$, it should be a much smaller value. And it's easy to see $M\geq 1$. Nevertheless, I cannot find any references and solutions to this.
And I learnt very little about functional analysis and calculus of variations too. I am just kind of interested in it derived from the above basic multivariable calculus problem and just wanna know the related results about it. Hence feel free to change the condition, or the dimension $n=2$. I got some links that may be related: Poincaré constant for a ball (circle), L2 norm of a function w.r.t. its gradient and Does a Lower Bounded Differentiable Function Always Have Arbitrarily Small Gradient?
Any references and help are appreciated. Thank you in advance!