Lions–Stampacchia theorem

Remarks

By the Riesz representation theorem there are a bounded linear operator $A$ on $H$ and a vector $f \in H$ with $a(u, v) = \ip{Au}{v}$ and $\ell(v) = \ip{f}{v}$. The inequality then says that $u = P_K\big(u - \rho (Au - f)\big)$ for any $\rho > 0$, where $P_K$ is the projection onto $K$ given by the Hilbert projection theorem. For $0 < \rho < 2\alpha / M^2$ the map $v \mapsto P_K\big(v - \rho (Av - f)\big)$ is a contraction of $K$, so the Banach fixed-point theorem gives existence and uniqueness. Uniqueness also follows directly: if $u_1, u_2$ are two solutions, adding the two inequalities gives $a(u_1 - u_2, u_1 - u_2) \le 0$, hence $u_1 = u_2$.

For $K = H$ the inequality becomes the equation $a(u, w) = \ell(w)$ for all $w \in H$, and the theorem reduces to the Lax–Milgram theorem. Example: the obstacle problem. Let $\Omega \subseteq \R^n$ be bounded and open, $H = H^1_0(\Omega)$, $a(u, v) = \int_\Omega \nabla u \cdot \nabla v \, dx$, and $K = \set{v \in H : v \ge \psi \text{ a.e.}}$ for an obstacle $\psi$ such that $K$ is nonempty. The theorem gives a unique $u \in K$ that minimizes the Dirichlet energy among functions lying above $\psi$. Coercivity cannot be dropped: for $H = \R$, $K = [0, \infty)$, $a = 0$ and $\ell(v) = v$, the inequality $0 \ge v - u$ for all $v \ge 0$ has no solution.

Going deeper

For a student

This is the basic existence theorem for variational inequalities and the constrained version of the Lax–Milgram theorem. In the symmetric case it is a statement about minimizing a strictly convex quadratic function over a closed convex set. The nonsymmetric case needs the fixed-point argument, since no energy is available. It applies to obstacle problems, to contact problems in elasticity such as the Signorini problem, and, after a change of unknown, to some free boundary problems.