I just came across the notation of the Lebesgue space $L^p_n$. However, I have never seen this before with the index $n$. Could anybody give a definition?
I suspect it has something to do with some kind of differentiability? Because in the paper (Screenshot) they do talk about $$div~q = 0 ~~~\text{ for } q \in L^p_n (Y)$$ for a (non-differentiable?) $q$, where $Y$ is the unit cube. No definition of $L^p_n$ is given.
Background: I am a student in applied math and need to understand the paper for a seminar. My advisor is absent for 2-3 weeks, but I still want to continue :)
I bet that this is the vector-valued $L^p(Y)$-space, so $L^p_n(Y) = (L^p(Y))^n$.
And the divergence is meant in a weak sense: $$\int_Y q^\top \, \nabla\varphi \, \mathrm{d}x = 0$$ for all $\varphi \in C_0^\infty(\hat Y)$, where $\hat Y$ is the interior of $Y$.
For functions $q \in H^1(Y)^n$, this is equivalent (integration by parts) to $\mathrm{div} q = 0$ (where $\mathrm{div}$ is the weak divergence), but generalizes to functions $q \in L^p(Y)^n$.