Page 112 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The vectorial case 99

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Theorem 3.19 Let n = N =2 and Ω ⊂ R be a bounded open set with Lipschitz
2
2
boundary. Let f : Ω×R ×R 2×2 −→ R, f = f (x, u, ξ),and F : Ω×R ×R 2×2 ×
R −→ R, F = F (x, u, ξ, δ), be continuous and satisfying
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f (x, u, ξ)= F (x, u, ξ, det ξ) , ∀ (x, u, ξ) ∈ Ω × R × R 2×2
where det ξ denotes the determinant of the matrix ξ. Assume also that

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(H1 vect ) (ξ, δ) → F (x, u, ξ, δ) is convex for every (x, u) ∈ Ω × R ;
(H2 vect )there exist p> max [q, 2] and α 1 > 0, α 2 ,α 3 ∈ R such that
p q 2 2×2
F (x, u, ξ, δ) ≥ α 1 |ξ| + α 2 |u| + α 3 , ∀ (x, u, ξ, δ) ∈ Ω × R × R × R .
Let u 0 ∈ W 1,p ¡ Ω; R 2 ¢ be such that I (u 0 ) < ∞, then (P) has at least one solu-
tion.
Remark 3.20 (i) It is clear that from the point of view of convexity the theorem
is more general than Theorem 3.3. Indeed if ξ → f (x, u, ξ) is convex then
choose F (x, u, ξ, δ)= f (x, u, ξ) and therefore (H1 vect ) and (H1) are equivalent.
However (H1 vect ) is more general since, for example, functions of the form

4 4
f (x, u, ξ)= |ξ| +(det ξ)
can be shown to be non convex, while
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F (x, u, ξ, δ)= |ξ| + δ
is obviously convex as a function of (ξ, δ).
(ii) The theorem is however slightly weaker from the point of view of coerciv-
ity. Indeed in (H2 vect )werequire p> 2, while in (H2) of Theorem 3.3 we only
asked that p> 1.
(iii) Similar statements and proofs hold for the general case n, N > 1.For
example when n = N =3 we ask that there exists a function
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F : Ω × R × R 3×3 × R 3×3 × R −→ R,F = F (x, u, ξ, η, δ)
so that

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f (x, u, ξ)= F (x, u, ξ, adj ξ, det ξ) , ∀ (x, u, ξ) ∈ Ω × R × R 3×3
2
where adj ξ denotes the matrix of cofactors of ξ (i.e. all the 2 × 2 minors of the
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matrix ξ). (H1 vect ) becomes then: (ξ, η, δ) → F (x, u, ξ, η, δ) is convex for every
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(x, u) ∈ Ω × R ;while (H2 vect ) should hold for p> max [q, 3].