[proofplan] Test the weak equation with $v=u$. Ellipticity controls the gradient, Poincare controls the $L^2$ part, and the dual norm of $f$ controls the right-hand side. [/proofplan]

[step:Test with the solution] Taking $v=u$ in the weak formulation gives \begin{align*} \int_U \sum_{i,j=1}^n A_{ij}\partial_{x_j}u\,\partial_{x_i}u\,d\mathcal L^n=f(u). \end{align*} Uniform ellipticity gives \begin{align*} \theta\|\nabla u\|_{L^2(U)}^2 \le f(u). \end{align*} By the definition of the $H^{-1}$ norm, \begin{align*} f(u)\le \|f\|_{H^{-1}(U)}\|u\|_{H_0^1(U)}. \end{align*} [/step]

[step:Convert gradient control to the full norm] Poincare's inequality on the bounded domain $U$ gives a constant $C_U$ such that \begin{align*} \|u\|_{H_0^1(U)}\le C_U'\|\nabla u\|_{L^2(U)} \end{align*} for some $C_U'>0$ depending only on $U$. Hence \begin{align*} \frac{\theta}{(C_U')^2}\|u\|_{H_0^1(U)}^2 \le \theta\|\nabla u\|_{L^2(U)}^2 \le \|f\|_{H^{-1}(U)}\|u\|_{H_0^1(U)}. \end{align*} If $u=0$ the estimate holds. Otherwise divide by $\|u\|_{H_0^1(U)}$ to obtain \begin{align*} \|u\|_{H_0^1(U)}\le \frac{(C_U')^2}{\theta}\|f\|_{H^{-1}(U)}. \end{align*} This gives the required estimate. [/step]