---
tag: H80M01T0H8
kind: statement
role: theorem
name: Lions–Stampacchia theorem
msc: [49J40, 47J20, 35J86]
uses: [HM2HHE3J51, 86KRHS6X22, 02AS45T3H9, 0ZCK8DQYN1, 4A82S5FVRT]
about: [6QS7X5GVMY]
sources:
  - url: https://en.wikipedia.org/wiki/Variational_inequality
---

## Statement

Let $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $K \subseteq H$ be a nonempty [[closed|0ZCK8DQYN1]] [[convex set|02AS45T3H9]].
Let $a \colon H \times H \to \R$ be a bounded [[coercive bilinear form|86KRHS6X22]], not necessarily symmetric: there are constants $M, \alpha > 0$ with $\abs{a(u, v)} \le M \norm{u} \norm{v}$ and $a(v, v) \ge \alpha \norm{v}^2$ for all $u, v \in H$.
Let $\ell$ be an element of the [[continuous dual space|4A82S5FVRT]] $H^*$.
Then there is exactly one $u \in K$ such that
$$a(u, v - u) \ge \ell(v - u) \quad \text{for all } v \in K.$$
If moreover $a$ is symmetric, then $u$ is the unique minimizer over $K$ of the function $v \mapsto \tfrac12 a(v, v) - \ell(v)$.

## 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|J5YNTHQMFX]].
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|85KK79FCN5]].
Example: the [[obstacle problem|GXN2NEHC8X]].
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.

## Explanations

### L1 human

Many equilibrium problems with a one-sided constraint, such as a membrane pushed against an obstacle, lead to an inequality rather than an equation.
The theorem says that, under natural conditions, such an inequality has exactly one solution.
The conditions are a closed convex set of allowed states and an energy-like form that grows at least quadratically.

### L2 human

This is the basic existence theorem for [[variational inequalities|6QS7X5GVMY]] 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.