For $f \in L^2(U)$ (U open and limited) consider the problem $-\Delta u = f$ em $U$ and $u=0 $ em $\partial U$. Let $J:H^1_0(U) \longrightarrow \mathbb{R}$ given by $$ J(u) = \frac{1}{2} \int_U |Du|^2 dx - \int_U fu \,dx. $$
(i) Show that $u \in H^1_0(U)$ it is a minimum of $J$, this is, $$ J(u)= \min_{v \in H^1_0(U)} J(v) $$ then $u$ is weak solution of the problem above.
(ii) Show that there is a constant $C > 0$ such that, for all $v \in H^1_0(U)$, $$ J(v) \geq - C \| f\|_{L^2(U)}. $$
(iii) Show that there is a constant $C_0 > 0$ such that, for all $v \in H^1_0(U)$, $$ \|v \|_{H^1(U)} \leq C_0( \| f\|_{L^2(U)} + J(v)). $$
(iv) Show that J has a minimum.
Obs: item (i) is the only one I managed to solve