= Solution
Take a minimizing sequence $(X_k,Y_k)$ in the affine Sobolev class $W$. The standing bounds $C^{-1}\leq X_k\leq C$ make the energy uniformly equivalent to
$$
\|\nabla X_k\|_2^2+\|\nabla Y_k\|_2^2.
$$
The fixed boundary values and the <Poincare inequality> therefore bound the sequence in $H^1(\Omega;\mathbb R^2)$. By weak compactness in a <reflexive Banach space>, a subsequence converges weakly to $(X,Y)$. The assumed weak closedness keeps the limit in $W$, and the assumed <weak lower semicontinuity> gives
$$
E[X,Y]\leq\liminf_kE[X_k,Y_k]=\inf_WE.
$$
Thus the <direct method in the calculus of variations> produces a minimizer. Its first variation vanishes in every compactly supported direction, so part (b) makes it a weak solution of the system.
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