In a vector field $F: \mathbb R^2 \ to \mathbb R^2$, can the Laplacian be non-zero over a surface (and not just a curve)? That is, in two dimensional ambient space, can there exist a two dimension region of non-zero Laplacian.
Intuitively, I think that it can not. The Laplacian is, intuitively, a measure of how much $F$ at a single point is greater or lesser than the average over the surrounding region. This can be formalized easily using integrals.
A single point can of course be greater than its surroundings: it is then a local maximum. And each point on a curve can be greater than the average of its surroundings: I visualize a ridge up the side of a mountain. Each point is of course not a maximum, but still has non-zero Laplacian.
But, visually, this cannot hold true for a contiguous two dimensional region. Am I correct? If so, how do we formalize this and prove it? If not: what is a counter example?
Update
GEdgar helped me find a simple counter example: $F(x,y) = (x^2, y^2)$. So I amend my original question: Can the Laplacian be non-zero over a bounded surface and be zero everywhere around that surface?
The field $F(x,y) = (x^2, y^2)$ gets away with non-zero Laplacian because it goes on forever, always getting greater than its average. But, if the field eventually has to be zero, this cannot go on forever.
So, let me refine my question: Can we have a field such that there exists a simple closed curve $\Gamma$ where the Laplacian is non-zero everywhere inside $\Gamma$ and zero everywhere outside (or at least on) $\Gamma$?