Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 341 / 4 / b / Solution

Define the energyIts first variation in direction $w\in \mathcal{H}$ isThus its minimizer $u$ satisfies the weak formulationwhich is the weak equation $Lu=f$. Positive definiteness makes $J$ strictly convex, so this stationary point is the unique minimizer; under uniform positivity of $p$, existence follows from the Lax-Milgram theorem.