Green-St Venant strain tensor is defined by $E(u)={1\over 2}[\nabla u+(\nabla u)^T+(\nabla u)^T\nabla u]$, where $\nabla u$ is the displacement gradient.

Show that

$u\in H^1(\Omega), E(u)\in L^r(\Omega), r\ge1\Rightarrow u\in W^{1,2r}(\Omega)$.

Here $H^1, W^{1,k}$ are standard Sobolev spaces, $\Omega$ is bounded domain in $R^3$.

It's an exercise(1.16) in P.G.Ciarlet's book "Mathematical Elasticity, Vol. I : Three-Dimensional Elasticity", cited as 'due to Luc Tartar'.

I don't have a clue on it. Any idea and/or comment are very much appreciated.