Download e-book for iPad: Direct Methods in the Calculus of Variations (Applied by Bernard Dacorogna

By Bernard Dacorogna

This publication is a brand new version of the authors past publication entitled Direct equipment within the Calculus of diversifications, 1989. it really is dedicated to the learn of vectorial difficulties within the calculus of diversifications. The ebook has been up-to-date considerably and a couple of extra examples were incorporated. The e-book will attraction researchers and graduate scholars in arithmetic and engineering.

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Example text

Furthermore, if f is convex, then levelα is convex for every α ∈ R. Part 3. Let fν : RN → R ∪ {+∞} , ν ∈ I, be an arbitrary family of convex (respectively lower semicontinuous) functions. Then f = sup fν ν∈I is a convex (respectively lower semicontinuous) function. 27 Note that in general the convexity of levelα f for every α ∈ R does not imply the convexity of f, as the following example indicates. Let f (x) = then 0 if x ≤ 0 1 if x > 0 ⎧ ∅ ⎪ ⎪ ⎨ (−∞, 0] levelα f = ⎪ ⎪ ⎩ R if α < 0 if 0 ≤ α < 1 if α ≥ 1 is convex for every α ∈ R, while f is not convex.

X ∈ Ω. 20). e. e. x ∈ Ω. 24). 18 Let Ω ⊂ Rn be a bounded open set, E ⊂ Rn and u0 be such that ∇u0 = ξ0 for some ξ0 ∈ Rn . e. x ∈ Ω, 20 Introduction then ξ0 ∈ E ∪ int co E. 3. 19 Let Ω ⊂ Rn be a bounded open set, 0 < a1 ≤ · · · ≤ an and ξ0 ∈ Rn×n be such that n n λi (ξ0 ) < i=ν i=ν ai , ν = 1, · · · , n. If u0 is an affine map such that ∇u0 = ξ0 , then there exists u ∈ u0 + W01,∞ (Ω; Rn ) so that, for almost every x ∈ Ω, λν (∇u (x)) = aν , ν = 1, · · · , n. 3 Some existence results for non-quasiconvex integrands We now apply (see Chapter 11) the results of the two previous sections to prove the existence of minimizers for (P ) inf I (u) = Ω f (∇u (x)) dx : u ∈ u0 + W01,∞ Ω; RN , where u0 is an affine map such that ∇u0 = ξ0 ∈ RN ×n , but without assuming any convexity or quasiconvexity hypothesis on the integrand f.

11 Let E ⊂ RN be convex and x ∈ ∂E. Then there exists a ∈ RN , a = 0, such that x; a ≤ x; a , for every x ∈ E. Proof. 6, we have that x ∈ ∂E. 10 (ii) to E, we have the claim. 3 Convex hull and Carath´ eodory theorem We start with the following definition. 12 The convex hull of a set E ⊂ RN , denoted by co E, is the smallest convex set containing E. 2, it is equivalent to say that co E is the intersection of all the convex sets that contain E. In the sequel we denote for any integer s s Λs := {λ = (λ1 , · · · , λs ) : λi ≥ 0 and i=1 λi = 1} .

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