 Solution

For every $v\in L^2(\Omega )$, the existence result in Question 2 applies to the identity coefficient matrix: $a_{ij}={\delta }_{ij}$ is bounded, symmetric and uniformly elliptic. It gives a unique $u\in W_0^{1,2}(\Omega )$ satisfyingThus every input has exactly one output and $\boxed{T\text{ is well defined}}$. The weak Dirichlet theorem used here requires only a bounded open set; no smoothness of its boundary is needed.