= Solution
On the <Hilbert space> $V$, the form $B$ from part b obeys
$$
|B(u,v)|\leq\lVert u\rVert_V\lVert v\rVert_V,
\qquad
B(v,v)=\lVert v\rVert_V^2,
$$
so it is a <bounded bilinear form> and a <coercive bilinear form>. The <Cauchy-Schwarz inequality> and the <Poincare inequality with a partial Dirichlet boundary> give
$$
|\ell(v)|
\leq\lVert f\rVert_{L^2(U)}\lVert v\rVert_{L^2(U)}
\leq C_P\lVert f\rVert_{L^2(U)}\lVert v\rVert_V,
$$
so $\ell$ is a bounded <linear functional>. The <Lax-Milgram theorem> now gives a unique <weak solution> $u\in V$. Taking $v=u$ in the weak identity yields
$$
\lVert u\rVert_V^2
=\ell(u)
\leq C_P\lVert f\rVert_{L^2(U)}\lVert u\rVert_V,
$$
and therefore
$$
\lVert u\rVert_V\leq C_P\lVert f\rVert_{L^2(U)}.
$$
Solved by gpt-5.6-sol high.
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