I have this execrise I've been trying to solve
Let $B=\{x \in \mathbb R^n: |x|<1\}, \quad a: \mathbb R^n \rightarrow \mathbb R , \quad a\in C^\infty(\mathbb R^n) \cap L^\infty(\mathbb R^n) $
$$\begin{cases} -\Delta u +a(\nabla u) = 1 & \text{in $B$} \\ \partial_{\nu }u + u = 0 & \text{in $\partial B$} \end{cases} $$
I've been able to show that the solution exists and it also belong to $H^2(B)$, the next point I'm trying to show is:
Show that every solution $u \in H^2 (B)$, is also $C^\infty( \bar B) $
In the solution of my professor the answer is:
Being $a (\nabla u) − 1 \in L^\infty (B)$ we get that $\nabla u ∈ L^\infty (B)$
Since $a \in C^\infty(B)$ and $ \nabla u \in H^1 (B;\mathbb R^n)$ we get $a(\nabla u) \in H^1(B)$
....
From now on is all clear to me, but how can I say that $\nabla u ∈ L^\infty (B)$ ??
Any help would be much appreciated