Weak Convergence Methods for Nonlinear Partial Differential ...

On the other hand, the div-curl lemma Corollary 2.64 shows that
Du (ν) : E(Du (ν) ) = ∑ ij
∂ j u (ν)
i E ij (Du (ν) ) ⇀ ∗ Du : E(Du)
since, for every i, curl ∇u i = 0 and div E i (Du (ν) ) = 0. So
This shows that
∫
Y : E(Y ) = Du : E.
(Y − Y ) : (E(Y ) − E(Y )) dν x (Y )
R m×n ∫ ∫
= Du : E − Y : E(Y ) dν x − Y dν x : E(Y ) + Y : E(Y )
= Y : E − Y : E − Y : E(Y ) + Y : E(Y ) = 0.
Noting that by assumption supp ν x is contained in a ball of radius ε 0 and that
E(Y ) = E(Y ) + DE(Y )(Y − Y ) + O(|Y − Y | 2 ),
we find
∫R m×n (Y − Y ) : DE(Y )(Y − Y ) dν x (Y ) ≤ Cε 0
∫
Now consider the function
R m×n |Y − Y | 2 dν x (Y ). (3.11)
F(x, P) := P : DE(Y )P − γ 2 |P |2 for P ∈ R m×n . (3.12)
By the strict Legendre-Hadamard condition (3.10) this function is quadratic and
rank-1-convex. But then Theorem 3.12 gives
∫
∫
F(x, Du(x)) dx ≤ lim inf F(x, Du ν ) dx.
ν→∞
U
for every U ⊂ Ω open. An approximation procedure similarly as in the proof of
Theorem 3.12 now shows that the term on the right hand side of this equality
can be expressed in terms of the Young measure, although F depends explicitly
on x: ∫
∫ ∫
lim F(Du ν ) dx = F(x, Y ) dν x (Y ) dx.
ν→∞
U
U R m×n
U being arbitrary, this proves that
∫
F(x, Du(x)) ≤ F(x, Y ) dν x (Y ) for a.e. x.
R m×n
73
U