Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup\_D |f|\leq1$.


Could someone calculate 
$$
\sup\limits\_{f \in \mathscr{H}} \inf\limits\_{z\in D} [f\_x^2(z)+f\_y^2(z)]\;?
$$


Current work:


I have an rigorous proof that it is less than 16.


Proof as follows:


Define $F(x,y)=f(x,y)+2(x^2+y^2)$. We have $F(0,0)\leq1$ and $F|\_{\partial D}\geq1$, which mean that minimun can be obtained in the interior. Therefore, we have an interior point $(x\_0,y\_0)$ s.t. 
$$F\_x(x\_0,y\_0)=F\_y(x\_0,y\_0)=0.
$$ 
Then $f\_x(x\_0,y\_0)=-4x\_0,f\_y(x\_0,y\_0)=-4y\_0$ $$f\_x^2(x\_0,y\_0)+f\_y^2(x\_0,y\_0)=16(x\_0^2+y\_0^2)<16.$$


Q.E.D.


Intuition:


However, the upper bound may be too rough. We could always find some point where $f\_x^2+f\_y^2$ is small. 


If there is some place slopes fast, then the restriction of $f$ means there is always somewhere plain.


Examples like $f(x,y)=x, y $ or $\displaystyle\frac{x+y}{\sqrt{2}}$ all have $f\_x^2+f\_y^2$ with 1.



$\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk
The supremum is indeed equal to $1$.


Suppose that $F$ is continuously differentiable and let $\Gamma$ be a maximal gradient line of $F$ in the unit disk (that is, $\Gamma$ is tangent to $\nabla F$ at each point of $\Gamma$, and the endpoints of $\Gamma$ either lie on the boundary of the disk or are critical points of $F$). Let $|\Gamma|$ be the length of $\Gamma$. Since $F$ takes values in $[-1, 1]$, by the mean value theorem there is a point $(x\_0,y\_0)$ on $\Gamma$ such that $|\nabla F(x\_0,y\_0)| \le 2 / |\Gamma|$.


If $F$ has a critical point, then there is $(x\_0, y\_0)$ such that $|\nabla F(x\_0, y\_0)| = 0$. Otherwise, all gradient lines begin and end at the boundary, and therefore there must be a gradient line of length at least $2$. It follows that there always is a point $(x\_0, y\_0)$ such that $|\nabla F(x\_0, y\_0)| \le 1$, as desired.


(This question seems to be more appropriate for Math.SE, though).


