 Solution

Differentiate under the integral and use the evolution equation:Thus $\boxed{I_f[u(t)]\le I_f[u(s)](t>s>0)}$. This is a gradient flow of the periodic screened Poisson energy. Part (iii) gives convergence in the function and its first derivative, so $\boxed{I_f[u(t)]\to I_f[u_f]}$, the minimum from part (ii).