Lax–Milgram theorem

The Lax–Milgram theorem guarantees a unique solution in a Hilbert space for every variational problem defined by a bounded, coercive bilinear form and a bounded linear functional.

Coercive bilinear form: A bilinear form whose value on a vector paired with itself is bounded below by a positive constant times its squared norm. Coercivity supplies the lower bound that prevents solutions from becoming unstable or disappearing.
Weak formulation: A reformulation of a differential equation in terms of integrals against test functions rather than pointwise derivatives. It expresses boundary-value problems in the variational form required by the theorem.
Bilinear form: A function of two vector arguments that is linear in each argument separately. The theorem places boundedness and coercivity conditions on this two-argument structure.
Fredholm alternative: A theorem relating solvability of a linear operator equation to the kernels of the operator and its adjoint. It handles broader operator equations, where coercivity and unique solvability may fail.
Bounded bilinear form: A bilinear form whose absolute value is bounded by a constant times the product of its arguments’ norms. Boundedness ensures the form defines a continuous operator on the Hilbert space.
Poisson equation: The elliptic partial differential equation stating that the Laplacian of a function equals a prescribed source. Its Dirichlet problem is a standard example whose weak solution follows from the theorem.
Linear functional: A linear map from a vector space to its scalar field. The variational equation’s forcing term is represented by a bounded linear functional.
Babuška–Lax–Milgram theorem: A generalization of Lax–Milgram giving well-posedness through inf-sup conditions for bilinear forms on paired spaces. It replaces coercivity on one Hilbert space with conditions suited to mixed formulations.
Riesz representation theorem: A theorem identifying every bounded linear functional on a Hilbert space with inner product against a unique vector. It converts the variational form into an operator on the Hilbert space.
Dirichlet boundary condition: A boundary condition prescribing the value of a solution on the boundary of a domain. For the Poisson problem, it helps define the function space and coercive energy form.

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