Let $\Box=[-1,1]^3$, $B_0=B(0,h)$ ($0 < h < 1)$, $E \in \mathcal{S}^0_3(\mathbb{R})$ a trace-free symetric matrix, $M \in \mathbb{R}$ fixed.
I am trying to find a function $\varphi$ such as :
\begin{equation} \left\{ \begin{aligned} \varphi \in (H^1(\Box))^3 \nonumber \\ \varphi|_{B_0} = - E x \nonumber \\ div \ \varphi = 0 \text{ on } \ \Box \setminus B_0\\ \int_{\Box} \varphi= M \nonumber \end{aligned} \right. \end{equation} where $(H^1(\Box))^3=\{w \in (L^2(\Omega))^3, \ \nabla w \in (L^2(\Omega))^9, \ \text{periodic conditions on } \Box \}$.
What I did so far :
I supposed I can find $\bar{\varphi}$ such as
\begin{equation} \left\{ \begin{aligned} \bar{\varphi} \in (H^1(\Box))^3 \nonumber \\ \bar{\varphi}|_{B_0} = - E x \nonumber \\ \end{aligned} \right. \end{equation} ( I think we can find such a $\bar{\varphi}$ that should be smooth since we don't ask too much ).
After that, I would like to use the Bogovskii operator Lemma (see after) to find a fonction $J \in (H^1_0(\Box \setminus B_0))^3$ such as \begin{equation} \left\{ \begin{aligned} div \ J= - div \ \bar{\varphi} \ \text{on} \ \Box \setminus B_0\\ J= 0 \ \text{on} \ \partial \Box\\ J= 0 \ \text{on} \ \partial B_0 \end{aligned} \right. \end{equation} which would be possible if I verify the compatibility condition : $$\int_{\Box \setminus B_0} div \ \bar{\varphi} = \int_{\partial \Box} \bar{\varphi} \cdot n + \int_{\partial B_0} \bar{\varphi} \cdot n =0$$ The first integral is indeed equals to zero thanks to the periodic boundary conditions, and the second one is also equals to zero : $\int_{\partial B_0} \bar{\varphi} \cdot n = -\int_{\partial B_0} Ex \cdot n = -\int_{ B_0} div \ (Ex) = \int_{B_0} Tr(E)=0$.
Then I extend $J$ to $0$ in $B_0$ and we pose $\varphi = J + \bar{\varphi} \in (H^1(\Box))^3$, which verifies
\begin{equation} \left\{ \begin{aligned} \varphi|_{B_0} = - E x \nonumber \\ div \ \varphi = div \ J + div \ \bar{\varphi} = 0 \text{ on } \ \Box \setminus B_0\\ \end{aligned} \right. \end{equation} which almost what I wanted (except for the mean of $\varphi ...$).
My questions
Do you think this is a good way of proceeding or should I use another method ?
How should I justify properly the existence of $\bar{\varphi}$ ?
Edit : I know now how to answer that question using a cut-off function $\chi$ that worths $1$ next to a neighborhood of $ \partial B_0$ and $0$ next to a neighborhood of $\partial \Box$. I then use the function $w : x \mapsto -Ex \in (H^1(B_0))^3$ that I extend in $\tilde{w} \in (H^1(\mathbb{R}^3))^3$ using the sobolev prolongation operator from $(H^1(B_0))^3$ to $(H^1(\mathbb{R}^3))^3$. Finally, I set $\bar{\varphi}=\chi \tilde{w}$ which should work.
Do you know how I could fix the mean of $\varphi$ to $M$ ?
Any help or advices are (as always !) welcome.
Bogovskii Operator Lemma
Let $\Omega$ a bounded regular domain of $\mathbb{R}^3$. Let $f \in (L^2(\Omega))^3$ such as
\begin{equation*} \int_{\Omega} f = 0. \end{equation*} Then there exists a fonction $v \in (H^1_0(\Omega))^3$ such as
\begin{equation*} div \ v =f \end{equation*} and there exists a constant $C$ such as $||\nabla v ||_{(L^2(\Omega))^9} \leq C ||f||_{(L^2(\Omega))^3}$.