  We provide new bounds on a flux integral over the portion of the boundary of
one regular domain contained inside a second regular domain, based on
properties of the second domain rather than the first one. This bound is
amenable to numerical computation of a flux through the boundary of a domain,
for example, when there is a large variation in the normal vector near a point.
We present applications of this result to occupational measures and
two-dimensional differential equations, including a new proof that all minimal
invariant sets in the plane are trivial.
