Critical Points Via Minimization

One of the most basic minimization problems one can pose is the following: Given a functional ϕ: E → ℝ on a Hilbert space E and a closed, convex subset C ⊂ E on which ϕ is bounded from below, find u 0 ∈ C such that $$ \phi \left( {u_0 } \right) = \mathop {\inf }\limits_{u \in C} \phi \left( u \right). $$ Of course, the problem as stated is much too general and one should be careful and make additional hypotheses! Nowadays, any good calculus student is aware of the fact that a function ϕ: ℝ → ℝ which is bounded from below on a closed interval C ⊂ ℝ does not necessarily attain its infimum in C. In fact, in learning the classic Weierstrass theorem, the student might have discovered that this delicate question of attaining the infimum is intimately connected to “continuity” of the functional ϕ and “compactness” of the set C! More precisely, the result that follows is well known and standard in a first course on Topology. For that, we recall that a functional ϕ: X → ℝ on a topological space X is lowersemicontinuous (l.s.c.) if ϕ−1(a, ∞) is open in X for any a ∈ ℝ (that is, ϕ−1(− ∞, a] is closed in X for any a ∈ ℝ). And if X satisfies the first countability axiom (for example, if X is a metric space), then ϕ: X → ℝ is l.s.c. if and only if ϕ(û) ≤ lim inf ϕ(un) for any û ∈ X and sequence u n converging to û.