# Inequality estimation

Let $B$ be an open unit ball in $\mathbb{R}^d$ centered at the origin and $u$ be a twice continuously differentiable function on $\bar{B}$ with $u|_{\partial B} = 0$. Know $$\Delta u = f.$$

How can I obtain such an inequality estimation($||\cdot||_2 = (\int_B |\cdot|^2 \, dx)^{1/2}$ indicates the $L^2$ norm): $$|| u||^2_2 + \sum_{i=1}^d || D_i u||^2_2 \leq 4||f ||^2_2,$$ where $D_i u :=\frac{\partial u}{\partial x^i}$.

The hint given in the book states both sides multiply with $(2-|x|^2)u$ and then integrate by parts, but I couldn't find out how to achieve the expected result.

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