Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 319 / 1 / c

Table of contents
- Solution c

Solution

Let

T = R / (2 π Z)

and define the positive self-adjoint operator

B = (1 - \partial_{x}^{2})^{1/2}

on

L^{2} (T)

. On the Fourier mode

e^{in x}

it acts by multiplication by

1 + n^{2}

. Consequently

D (B) = H^{1} (T), D (B^{2}) = H^{2} (T) .

The energy space is the periodic Sobolev space

H = H^{1} (T) \times L^{2} (T),

with inner product

((u, v), (p, q))_{H} = (B u, Bp)_{L^{2}} + (v, q)_{L^{2}} .

If

u = \sum u_{n} e^{in x}

and

v = \sum v_{n} e^{in x}

, then

∥ (u, v) ∥_{H}^{2} = 2 π \sum_{n \in Z} [(1 + n^{2}) ∣ u_{n} ∣^{2} + ∣ v_{n} ∣^{2}],

which is precisely the stated energy norm.

With

v = u_{t}

, the periodic Klein-Gordon equation becomes

\dot{Z} = A Z, A = (0 - B^{2} I 0), A (u, v) = (v, - B^{2} u) .

For

A Z

to belong to

H^{1} \times L^{2}

, one needs

v \in H^{1}

and

u \in H^{2}

. Thus

D (A) = H^{2} (T) \times H^{1} (T) .

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