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1997 VJIMC, Problem 3

Let $u\in C^2(\overline D)$, $u=0$ on $\partial D$ where $D$ is the open unit ball in $\mathbb R^3$. Prove that the following inequality holds for all $\varepsilon>0$: $$\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.$$(We recall that $\nabla u$ and $\Delta u$ are the gradient and Laplacian, respectively.)

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