Legendre-Fenchel duality in elasticity

We show that the displacement and strain formulations of the displacement-traction problem of three-dimensional linearized elasticity can be viewed as Legendre-Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre-Fenchel duality. Résumé Dualité de Legendre-Fenchel en élasticité. On montre que les formulations en déplacements et en déformations du problème en déplacement-traction de l’élasticité linéarisée tri-dimensionnelle peuvent être vues comme des problèmes duaux de Legendre-Fenchel de la formulation en contraintes de ce même problème. On montre également que chacun des Lagrangiens correspondants a un point-selle, justifiant ainsi complètement cette nouvelle approche de l’élasticité au moyen de la dualité de Legendre-Fenchel. 1. Legendre-Fenchel duality All vector spaces, matrices, etc., considered in this Note are real. The dual space of a normed vector space X is denoted X, and X∗〈·, ·〉X designates the associated duality. The bidual space of X is denoted X; if X is a reflexive Banach space, X will be identified with X by means of the usual canonical isometry. The indicator function IA of a subset A of a set X is the function IA defined by IA(x) := 0 if x ∈ A and IA(x) := +∞ if x / ∈ A. A function g : X → R ∪ {+∞} is proper if {x ∈ X ; g(x) < +∞} 6 = ∅. Let Σ be a normed vector space and let g : Σ → R ∪ {+∞} be a proper function. The Legendre-Fenchel transform of g is the function g : Σ → R ∪ {+∞} defined by g : e ∈ Σ → g(e) := sup σ∈Σ {Σ∗〈e, σ〉Σ − g(σ)}. The next theorem summarizes some basic properties of the Legendre-Fenchel transform when the space Σ is a reflexive Banach space. For proofs, see, e.g., Ekeland & Temam [7] or Brezis [2]. The equality g = g constitutes the Fenchel-Moreau theorem. Email addresses: [email protected] (Philippe G. Ciarlet), [email protected] (Giuseppe Geymonat), [email protected] (Francoise Krasucki). Preprint submitted to the Académie des sciences March 4, 2011 Theorem 1.1 Let Σ be a reflexive Banach space, and let g : Σ → R∪{+∞} be a proper, convex, and lower semi-continuous function. Then the Legendre-Fenchel transform g : Σ → R ∪ {+∞} of g is also proper, convex, and lower semi-continuous. Let g : σ ∈ Σ → g(σ) := sup e∈Σ {Σ∗〈e, σ〉Σ − g (e)} denote the Legendre-Fenchel transform of g (recall that X is here identified with X). Then g = g. Given a minimization problem infσ∈Σ G(σ), called (P), with a function G : Σ → R∪{+∞} of the specific form given in Theorem 1.2 below, the following simple result will be the basis for defining two different dual problems of problem (P). The functions L and L̃ defined in the next theorem are the Lagrangians associated with the minimization problem (P). Theorem 1.2 Let Σ and V be two reflexive Banach spaces, let g : Σ → R∪{+∞} and h : V ∗ → R∪{+∞} be two proper, convex, and lower semi-continuous functions, let Λ : Σ → V ∗ be a linear and continuous mapping, let the function G : Σ → R ∪ {+∞} be defined by G : σ ∈ Σ → G(σ) := g(σ) + h(Λσ), and finally, let the two functions L : Σ× Σ → {−∞} ∪ R ∪ {+∞} and L̃ : Σ× V → {−∞} ∪ R ∪ {+∞} be defined by L : (σ, e) ∈ Σ× Σ → L(σ, e) := Σ∗〈e, σ〉Σ − g (e) + h(Λσ), L̃ : (σ, v) ∈ Σ× V → L̃(σ, v) := g(σ) + V ∗〈Λσ, v〉V − h (v). Then inf σ∈Σ G(σ) = inf σ∈Σ sup e∈Σ L(σ, e) = inf σ∈Σ sup v∈V L̃(σ, v). A key issue then consists in deciding whether the infimum found in problem (P) is equal to the supremum found in either one of its dual problems, i.e., for instance in the case of the first dual problem (to fix ideas), whether inf σ∈Σ G(σ) = sup e∈Σ G(e), or equivalently, inf σ∈Σ sup e∈Σ L(σ, e) = sup e∈Σ inf σ∈Σ L(σ, e). If this is the case, the next issue consists in deciding whether the Lagrangian L possesses a saddle-point (σ, e) ∈ Σ× Σ, i.e., that satisfies inf σ∈Σ sup e∈Σ L(σ, e) = inf σ∈Σ L(σ, e) = L(σ, e) = sup e∈Σ L(σ, e) = sup e∈Σ inf σ∈Σ L(σ, e). This is precisely the type of questions addressed in this Note, the point of departure (P) being a classical quadratic minimization problem arising in three-dimensional linearized elasticity. 2. Functional analytic preliminaries Latin indices vary in the set {1, 2, 3}, save when they are used for indexing sequences, and the summation convention with respect to repeated indices is systematically used in conjunction with this rule. A domain in R is a bounded, connected, open subset of R whose boundary, denoted Γ, is Lipschitzcontinuous, the set Ω being locally on a single side of Γ. Spaces of functions, vector fields in R, and 3 × 3 symmetric matrix fields, defined over an open subset of R are respectively denoted by italic capitals, boldface Roman capitals, and special Roman capitals. The inner-product of a ∈ R and b ∈ R is denoted a · b. The notation s : t := sijtij designates the matrix inner-product of two matrices s := (sij) and t := (tij) of order three. The inner product in the space L(Ω) is given by (σ, τ ) ∈ L(Ω) × L(Ω) → ∫ Ω σ : τ dx, and ‖·‖L2(Ω) denotes the corresponding norm. The space L(Ω) will be identified with its dual space. The duality bracket between the space H(Γ) and its dual space will be denoted 〈·, ·〉Γ := H−1/2(Γ)〈·, ·〉H1/2(Γ). 2 For any vector field v = (vi) ∈ D (Ω), the associated linearized strain tensor is the symmetric matrix field ∇sv ∈ D (Ω) defined by ∇sv := 1 2 (∇v T +∇v). We now recall some functional analytic preliminaries, due to Geymonat & Suquet [10] and Geymonat & Krasucki [8,9]. Given a domain Ω in R, define the space H(div; Ω) := {μ ∈ L(Ω); divμ ∈ L(Ω)}. The set Ω being a domain, the density of the space C(Ω) in the space H(div; Ω) then implies that the mapping μ ∈ C(Ω) → μν|Γ can be extended to a continuous linear mapping from the space H(div; Ω) into H(Γ), which for convenience will be simply denoted μ ∈ H(div; Ω) → μν ∈ H(Γ). Theorem 2.1 The Green formula ∫