Almost invariant rectifiable currents of codimension 1 for field line flows
Felicia Bernatzki
Abstract:
Let $B \in C^{0}(\RA, \RA)$ be a vectorfield and
let $\T \subset \subset \RA$ be an $m+1$-dimensional manifold with boundary.
To analyze the field line structure of $B$ in $\T$ it is useful
to find a family of nested hypersurfaces to which $B$ is tangential.
However, when $B$ has a chaotic field
line structure one cannot find such a family.
Instead, here we show
the existence of a nested continuous family of
$m$-rectifiable currents in $\T$ $(S_{t})_{t \in [0,1]}$ which we obtain from
minimizers of a functional $E_{\epsilon}(B)(\cdot)$ ($\epsilon>0$
is an arbitrarily small parameter). The form of the functional $E_{0}(B)$ is motivated
by problems arising
in the context of plasma confinement.
For an embedded hypersurface
$S \subset \bf{R}^{m+1}$
this functional is given by
$$E_{\epsilon}(B)(S):=
\int_{S} \langle B, n^{S}\rangle ^{2} d {\cal{H}}^{m}
+ \varphi(\epsilon( {\cal{H}}^{m}(S)+ \int_{S}|H^{S}|^{p} d {\cal{H}}^{m})),$$
where $n^{S}$ is a unit normal vector of $S$, $H^{S}$ is its mean
curvature and $ \varphi$ is a monotone increasing function.
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