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{c}
{title2=$\|u\|_3\leq C\|u\|_2^{1/2}\|u\|_{H^1}^{1/2}$}

On a bounded three-dimensional domain, interpolation between $L^2$ and the <Sobolev inequality> $H^1\hookrightarrow L^6$ gives
$$
\|u\|_{L^3}\leq C\|u\|_{L^2}^{1/2}\|u\|_{H^1}^{1/2}.
$$

H1 L3 interpolation inequality in three dimensions

Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 105 / 2 / b / i / Solution

H1 L3 interpolation inequality in three dimensions ( $∥ u ∥_{3} \leq C ∥ u ∥_{2}^{1/2} ∥ u ∥_{H^{1}}^{1/2}$ )

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H1 L3 interpolation inequality in three dimensions (∥u∥3​≤C∥u∥21/2​∥u∥H11/2​)

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H1 L3 interpolation inequality in three dimensions ( $∥ u ∥_{3} \leq C ∥ u ∥_{2}^{1/2} ∥ u ∥_{H^{1}}^{1/2}$ )

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