5 - Critical points: extrema

Summary

In this chapter we will discuss the existence of maxima and minima for a functional on a Hilbert or Banach space.

Functionals and critical points

Let E be a Banach space. A functional on E is a continuous real valued map J : E → ℝ.

More in general, one could consider functionals defined on open subsets of E. But, for the sake of simplicity, in the sequel we will always deal with functionals defined on all of E, unless explicitly remarked.

Let J be (Fréchet) differentiable at u ∈ E with derivative dJ(u) ∈ L(E, ℝ). Recall that (see Section 1.1):

• if J is differentiable on E, namely at every point u ∈ E and the map E ↦ → L(E, ℝ), u ↦ dJ(u), is continuous, we say that J ∈ C¹(E,ℝ);

• if J is k times differentiable on E with kth derivative d^kJ(u) ∈ L_k(E,ℝ) (the space of k-linear maps from E to ℝ) and the application E ↦ L_k(E,ℝ), u ↦ d^kJ(u), is continuous, we say that J ∈ C^k(E,ℝ).

Definition 5.1A critical, or stationary, point of J : E → ℝ is a z ∈ E such that J is differentiable at z and dJ(z) = 0. A critical level of J is a number c ∈ ℝ such that there exists a critical point z ∈ E with J(z) = c. The set of critical points of J will be denoted by Z, while Z_cwill indicate the set of critical points at level c: Z_c = {z ∈ Z : J(z) = c}.