= Solution
We prove existence directly through <quadratic energy minimization by the parallelogram law>. Let $m=\inf_{v\in H_g^1}A[v]$. The set contains $g$, and <uniform ellipticity> gives $0\leq m\leq A[g]<\infty$. Choose a <minimizing sequence> $v_j$ with $A[v_j]\to m$. Since $(v_j+v_k)/2\in H_g^1$, the <parallelogram law> gives
$$
A[v_j-v_k]=2A[v_j]+2A[v_k]-4A[(v_j+v_k)/2]\leq2A[v_j]+2A[v_k]-4m\longrightarrow0.
$$
The differences belong to $H_0^1$, so <uniform ellipticity> and the <Poincaré inequality> imply
$$
\|v_j-v_k\|_{H^1}^2\leq\frac{1+C_P^2}{\lambda}A[v_j-v_k]\longrightarrow0.
$$
Thus $(v_j)$ is a <Cauchy sequence>. Completeness of $H_0^1$, applied to $v_j-g$, produces $u\in H_g^1$ with $v_j\to u$ in $H^1$. The bounded coefficient matrix makes $a$ a <bounded bilinear form>, so $A[v_j]\to A[u]$. Consequently $A[u]=m$, and part (i) makes $u$ a <weak solution>.
For uniqueness, the difference $w$ of two <weak solutions> lies in $H_0^1$ and satisfies $a(w,w)=0$. <Uniform ellipticity> gives $\nabla w=0$ almost everywhere, and the <Poincaré inequality> gives $w=0$. The argument proves existence and uniqueness without assuming the <Lax-Milgram theorem>.
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