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

Table of contents
- Solution i

Solution

The Holder inequality interpolates between

L^{2} (U)

and

L^{6} (U)

:

∥ v ∥_{L^{3}}^{3} = \int_{U} ∣ v ∣^{3/2} ∣ v ∣^{3/2} \leq ∥ v ∥_{L^{2}}^{3/2} ∥ v ∥_{L^{6}}^{3/2} .

Taking cube roots and applying the three-dimensional Sobolev inequality

H^{1} (U) ↪ L^{6} (U)

gives

∥ v ∥_{L^{3} (U)} \leq C ∥ v ∥_{L^{2} (U)}^{1/2} ∥ v ∥_{H^{1} (U)}^{1/2} .

This is the H1 L3 interpolation inequality in three dimensions.

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