= Solution
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>.
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