LaxMilgram

The Lax–Milgram theorem is a foundational result in functional analysis and the theory of partial differential equations. It guarantees existence and uniqueness of solutions to linear variational problems in a Hilbert space setting.

Let V be a real or complex Hilbert space, a: V×V → F be a continuous bilinear form, and suppose a is V-coercive, meaning there exists α > 0 with a(v, v) ≥ α ||v||^2 for all v ∈ V. For every f in the dual space V', there exists a unique u ∈ V such that a(u, v) = f(v) for all v ∈ V. Moreover, the solution depends continuously on f, with the stability estimate ||u|| ≤ (1/α) ||f||.

Equivalently, the map T: V → V' defined by T(u) = a(u, ·) is a continuous linear isomorphism onto V'. The coercivity of a ensures that T is invertible, so the variational problem is well posed.

Applications are widespread in the study of linear elliptic boundary-value problems. In the standard weak formulation of Poisson’s equation, for example, one seeks u ∈ H0^1(Ω) such that ∫Ω ∇u · ∇v dx = ∫Ω f v dx for all v ∈ H0^1(Ω). Here a(u, v) = ∫Ω ∇u · ∇v dx is continuous and coercive on H0^1(Ω) by Poincaré’s inequality, guaranteeing a unique weak solution for appropriate f.

The theorem applies to real and complex Hilbert spaces and underpins many numerical methods, including finite element analysis, by ensuring well-posedness of both continuous and discrete variational problems.