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 Incoming links: Gradient flow of the periodic screened Poisson energy

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 29C / iv / Solution Created 2026-09-24 Updated 2026-10-14

Differentiate under the integral and use the evolution equation:

\frac{d}{d t} I_{f} [u (t)] = \int_{- π}^{π} (- u_{xx} + u - f) u_{t} d x = - \int_{- π}^{π} u_{t}^{2} d x \leq 0.

Thus

I_{f} [u (t)] \leq I_{f} [u (s)] (t > s > 0)

. This is a gradient flow of the periodic screened Poisson energy. Part (iii) gives convergence in the function and its first derivative, so

I_{f} [u (t)] \to I_{f} [u_{f}]

, the minimum from part (ii).

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