Critical Points under Constraints

Abstract

In many variational problems one must find critical points of a given functional ϕ ∈ C¹(X, ℝ) in the presence of constraints, that is, critical points of ϕ restricted to a set M ⊂ X of constraints. Naturally, in order to be able to talk about critical points of ϕ|M, the set M must have a differentiable structure. Typically, in the case of a finite number of constraints, M is of the form M = {u ∈ X | Ψ (u) = 0, j = 1,..., k} where Ψ_j ∈ C¹(X, ℝ), j = 1,...,k.