Lax-Milgram Theorem

Core Idea

The Lax-Milgram theorem guarantees the existence and uniqueness of solutions to abstract variational problems: if a bilinear form a(u,v) on a Hilbert space H is continuous and coercive (a(u,u) ≥ α||u||² for some α > 0), then for every continuous linear functional F on H, there exists a unique u ∈ H with a(u,v) = F(v) for all v ∈ H. This abstract result, when applied to the weak formulations of elliptic PDEs, immediately yields existence and uniqueness of weak solutions. It does not require symmetry of the bilinear form, making it more general than the Riesz representation theorem or variational minimization.

Explainer

The Lax-Milgram theorem is the workhorse existence result for elliptic PDEs. It reduces the question of solvability for a PDE to verifying two functional-analytic properties of a bilinear form: continuity and coercivity. Once these are established—which is typically a matter of applying standard inequalities like Poincare, Cauchy-Schwarz, and trace inequalities—existence, uniqueness, and continuous dependence all follow immediately from the abstract theorem.

The theorem generalizes the Riesz representation theorem, which handles only the case where a(u,v) = (u,v)_H is the inner product itself. When a is symmetric and coercive, the problem a(u,v) = F(v) is equivalent to minimizing J(v) = ½a(v,v) - F(v), and existence follows from the direct method of the calculus of variations. But many important PDEs—convection-diffusion equations -Δu + b·∇u = f, for example—have non-symmetric weak formulations, and the Lax-Milgram theorem handles these directly.

The proof is elegant and short. Define A: H → H by (Au, v) = a(u,v) for all v (possible by Riesz). Continuity of a means A is bounded: ||Au|| ≤ M||u||. Coercivity means A is bounded below: α||u||² ≤ a(u,u) = (Au, u) ≤ ||Au|| ||u||, so ||Au|| ≥ α||u||. This shows A is injective with closed range. A brief argument (or application of the closed range theorem) shows the range is all of H, so A is an isomorphism and u = A⁻¹(RF) where R is the Riesz map.

In applications, verifying the hypotheses of Lax-Milgram for specific PDEs is a systematic exercise. For the diffusion equation -div(a(x)∇u) = f with a(x) ≥ a₀ > 0 (uniformly elliptic), the bilinear form a(u,v) = ∫a(x)∇u·∇v dx is coercive on H¹₀ by ellipticity and the Poincare inequality, and continuous by the bound a ≤ a₁. For the convection-diffusion equation with additional terms ∫b·∇u·v dx + ∫cu·v dx, one verifies that the lower-order terms do not destroy coercivity (which holds when c - ½div b ≥ 0, or when the lower-order terms are dominated by the diffusion). The Lax-Milgram theorem is also the theoretical foundation for the finite element method, where the Galerkin approximation inherits the same existence and stability properties from the continuous problem.