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.



A property on the Green-St Venant strain tensor
Answer given by pde\_bk in comments:


Consider the diagonal entries. For each $i=1,2,3$,
$$
E\_{ii}(u) = \partial\_i u\_i + \frac{1}{2}\sum\_{k=1}^3 \left|\partial\_i u\_k\right|^{2} \quad\quad (\star).
$$
Suppose first $1\leq r \leq 2$. As $u\in W^{1,2}$, $\partial\_i u\_i \in L^r$.


Thus if $E\_{ii} \in L^{r}$ for each $i=1,2,3$ then 
$$\sum\_{i=1}^3\sum\_{k=1}^3 \left|\partial\_i u\_k\right|^2 = \left|\nabla u\right|^2 \in L^r, \quad\quad(\star\star)$$
 and therefore $u\in W^{1,2r}$.


By induction suppose now $E\in L^r$ with $2^{n}<r\leq2^{n+1}$ (we just did $n=0$), and that the result holds for $r=2^n$. 
Since $E\in L^{r}\subset L^{2^{n}}$, we have from the induction hypothesis $\nabla u \in L^{2^{n+1}}$, and therefore $\nabla u \in L^{r}$. Using ($\star$) and ($\star\star$), the conclusion follows. 


