Book contents
- Frontmatter
- Contents
- Preface
- 1 Preliminaries
- Part I Topological methods
- Part II Variational methods, I
- 5 Critical points: extrema
- 6 Constrained critical points
- 7 Deformations and the Palais–Smale condition
- 8 Saddle points and min-max methods
- Part III Variational methods, II
- Part IV Appendices
- References
- Index
5 - Critical points: extrema
from Part II - Variational methods, I
Published online by Cambridge University Press: 19 May 2010
- Frontmatter
- Contents
- Preface
- 1 Preliminaries
- Part I Topological methods
- Part II Variational methods, I
- 5 Critical points: extrema
- 6 Constrained critical points
- 7 Deformations and the Palais–Smale condition
- 8 Saddle points and min-max methods
- Part III Variational methods, II
- Part IV Appendices
- References
- Index
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 dkJ(u) ∈ Lk(E,ℝ) (the space of k-linear maps from E to ℝ) and the application E ↦ Lk(E,ℝ), u ↦ dkJ(u), is continuous, we say that J ∈ Ck(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 Zcwill indicate the set of critical points at level c: Zc = {z ∈ Z : J(z) = c}.
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- Information
- Nonlinear Analysis and Semilinear Elliptic Problems , pp. 77 - 88Publisher: Cambridge University PressPrint publication year: 2007