Define the energy
$$
J(v)=\frac12a(v,v)-\int_{-1}^1f(x)v(x)dx.
$$
Its first variation in direction $w\in\mathcal H$ is
$$
\delta J(v;w)=a(v,w)-\int_{-1}^1fw\,dx.
$$
Thus its minimizer $u$ satisfies the <weak formulation>
$$
a(u,w)=\int_{-1}^1fw\,dx
\qquad\text{for every }w\in\mathcal H,
$$
which 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>.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/iii/paper-341/4/b/solution