= 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 $u\in W_0^{1,2}(\Omega)$ satisfying
$$
\int_\Omega Du\cdot D\phi=-\int_\Omega v\phi\qquad(\phi\in W_0^{1,2}(\Omega)).
$$
Thus 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.
Back to article page
