Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 4 / 23H / c

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- Solution c

Solution

The zero-boundary Sobolev space is

H_{0}^{1} (Ω) = \overline{C_{c}^{\infty} (Ω)}^{H^{1} (Ω)} .

Here

H^{1} (Ω)

consists of the

L^{2}

functions whose first weak derivatives also belong to

L^{2}

, with norm

∥ u ∥_{H^{1}}^{2} = \int_{Ω} (∣ u ∣^{2} + ∣\nabla u ∣_{2}^{2}) .

On a sufficiently regular domain, the closure condition is equivalent to having zero boundary trace.

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