In a homework question, we've been given the following things:
Let $B=\overline{B(0,1)} \subseteq \mathbb R^n$ be the closed unit ball, $f : B\to \mathbb R^n$ a $\mathcal C^2$ function and let some $a_i : B \to \mathbb R$ be continuous for $i=1,\ldots,n$.
For all $x\in B$, we have $\Delta f(x)+\sum_{i=1}^n a_i(x)\frac{\partial f}{\partial x_i}(x) > 0$ where $\Delta f$ is the Laplacian of $f$.
Then show that $f$ has no local maximum in $Int(B)$.
I'm kind of confused by the question. We're used to having questions where all the premises matter, but this doesn't seem to be the case here, or is it?
I mean, let $x_0$ be the maximum whose existence we want to contradict. Then $x_0$ has to be a critical point and the sum in the inequality vanishes anyway. But $\Delta f$ needs to be negative in maxima, right? Hence contradiction.
So, what's the point of the $a_i$, why make them continuous? Do we really need the contraint on the unit ball, or if not, what property of $B$ matters? Convexity, compactness?
Are all these premises just to confuse the students and we actually don't need them? Or am I severely overlooking something? If so, could you give a hint please on what to keep in mind for the proof?
Thank you.