Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $|\cdot|$ its induced norm. A bilinear form $[\cdot, \cdot]:H \times H \to \mathbb R$ is said to be
- continuous if there is a constant $C>0$ such that $$ |[u, v]| \le C |u| |v| \quad \forall u,v \in H. $$
- coercive if there is a constant $\alpha>0$ such that $$ [v, v] \ge \alpha |v|^2 \quad \forall v \in H. $$
Then we have
Stampacchia theorem Assume that $[\cdot, \cdot]:H \times H \to \mathbb R$ is a continuous coercive bilinear form. Let $K$ be a non-empty closed convex subset of $H$. Then for each $\varphi \in H^*$, there is a unique $u\in K$ such that $$ [u, v-u] \ge \varphi (v-u) \quad \forall v\in K. $$ Moreover, if $[\cdot, \cdot]$ is symmetric, then $u$ is characterized as a minimizer of $$ \min_{v\in K} \bigg \{ \frac{1}{2} [v, v] - \varphi (v) \bigg \}. $$
Lax-Milgram theorem Assume that $[\cdot, \cdot]:H \times H \to \mathbb R$ is a continuous coercive bilinear form. Then for each $\varphi \in H^*$, there is a unique $u\in H$ such that $$ [u, v] = \varphi (v) \quad \forall v\in H. $$ Moreover, if $[\cdot, \cdot]$ is symmetric, then $u$ is characterized as a minimizer of $$ \min_{v\in H} \bigg \{ \frac{1}{2} [v, v] - \varphi (v) \bigg \}. $$
It's straightforward to prove Lax-Milgram theorem by Stampacchia theorem.
Can we prove Stampacchia theorem by Lax-Milgram theorem?