In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant under the flow defined by a vector field, namely at each point of the closed set the vector field must have non-positive inner product with any exterior normal vector to the set.
The generalization to non-smooth subsets is important in the theory of partial differential equations.
[1] Let F be closed subset of a C2 manifold M and let X be a vector field on M which is Lipschitz continuous.
To prove the reverse implication, since the result is local, it enough to check it in Rn.
In that case X locally satisfies a Lipschitz condition If F is closed, the distance function D(x) = d(x,F)2 has the following differentiability property: where the minimum is taken over the closest points z to x in F. The differentiability property implies that minimized over closest points z to c(t).