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integral inequality in R^3

Source: VJIMC 1997 2.3

October 9, 2021
calculusintegrationinequalitiesMultivariable Calculus

Problem Statement

Let u∈C2(D‾)u\in C^2(\overline D), u=0u=0 on ∂D\partial D where DD is the open unit ball in R3\mathbb R^3. Prove that the following inequality holds for all ε>0\varepsilon>0: ∫D∣∇u∣2dV≤ε∫D(Δu)2dV+14ε∫Du2dV.\int_D|\nabla u|^2dV\le\varepsilon\int_D(\Delta u)^2dV+\frac1{4\varepsilon}\int_Du^2dV.(We recall that ∇u\nabla u and Δu\Delta u are the gradient and Laplacian, respectively.)
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