I have a question about quasi-convexity or more precisely if I change the boundary condition on the test function. More precisely, let say I have a function $W: \mathbb{R}^{d\times d}\rightarrow (0,\infty)$ such that for every $\Lambda\in \mathbb{R}^{d\times d}$ it holds $$W(\Lambda)\leq \frac{1}{\vert B\vert}\int_{B}W(\Lambda+\nabla\phi)\quad\text{for any $\phi\in W^{1,\infty}(B)$ such that $\int_{B}\nabla\phi=0$},$$ i.e I have Neumann boundary conditions.
Does it imply that $W$ is quasi-convex (in the sense of Morrey) ? If not, what kind of function satisfies the condition above ?
Thanks in advance !