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  "body": "\n## Statement\n\nLet $H$ be a real [[Hilbert space|HM2HHE3J51]] and let $K \\subseteq H$ be a nonempty [[closed|0ZCK8DQYN1]] [[convex set|02AS45T3H9]].\nLet $a \\colon H \\times H \\to \\R$ be a bounded [[coercive bilinear form|86KRHS6X22]], not necessarily symmetric: there are constants $M, \\alpha \u003e 0$ with $\\abs{a(u, v)} \\le M \\norm{u} \\norm{v}$ and $a(v, v) \\ge \\alpha \\norm{v}^2$ for all $u, v \\in H$.\nLet $\\ell$ be an element of the [[continuous dual space|4A82S5FVRT]] $H^*$.\nThen there is exactly one $u \\in K$ such that\n$$a(u, v - u) \\ge \\ell(v - u) \\quad \\text{for all } v \\in K.$$\nIf moreover $a$ is symmetric, then $u$ is the unique minimizer over $K$ of the function $v \\mapsto \\tfrac12 a(v, v) - \\ell(v)$.\n\n## Remarks\n\nBy 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 $\\ell(v) = \\ip{f}{v}$.\nThe inequality then says that $u = P_K\\big(u - \\rho (Au - f)\\big)$ for any $\\rho \u003e 0$, where $P_K$ is the projection onto $K$ given by the [[Hilbert projection theorem|J5YNTHQMFX]].\nFor $0 \u003c \\rho \u003c 2\\alpha / M^2$ 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.\nUniqueness 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 $u_1 = u_2$.\n\nFor $K = H$ 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]].\nExample: the [[obstacle problem|GXN2NEHC8X]].\nLet $\\Omega \\subseteq \\R^n$ 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.\nThe theorem gives a unique $u \\in K$ that minimizes the Dirichlet energy among functions lying above $\\psi$.\nCoercivity 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.\n\n## Explanations\n\n### L1 human\n\nMany equilibrium problems with a one-sided constraint, such as a membrane pushed against an obstacle, lead to an inequality rather than an equation.\nThe theorem says that, under natural conditions, such an inequality has exactly one solution.\nThe conditions are a closed convex set of allowed states and an energy-like form that grows at least quadratically.\n\n### L2 human\n\nThis is the basic existence theorem for [[variational inequalities|6QS7X5GVMY]] and the constrained version of the Lax–Milgram theorem.\nIn the symmetric case it is a statement about minimizing a strictly convex quadratic function over a closed convex set.\nThe nonsymmetric case needs the fixed-point argument, since no energy is available.\nIt 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.\n",
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