Denote $f:\mathbb{R^2}\rightarrow\mathbb{R}$ has a continuous twice derivative on the disc $D=\{(x,y):x^2+y^2\leq1\}$, which satisfy $f(\partial D)=\{0\} $, where $\partial D=\{(x,y):x^2+y^2=1\}$, please prove:
$$\iint_Df(x,y)(\frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2})dxdy\leq0$$
I tried using the polar coordinate, but can't put one foot forward.