Abstract:
We show that every isoperimetric set in $\R^N$ with density is
bounded if the density is continuous and bounded by above and below.
This improves the previously known boundedness results, which
basically needed a Lipschitz assumption; on the other hand, the
present assumption is sharp, as we show with an explicit example. To
obtain our result, we observe that the main tool which is often used,
namely a classical “$\epsilon-\epsilon$” property already discussed
by Allard, Almgren and Bombieri, admits a weaker counterpart which is
still sufficient for the boundedness, namely, an
“$\epsilon-\epsilon^\beta$” version of the property. And in turn,
while for the validity of the first property the Lipschitz assumption
is essential, for the latter the sole continuity is enough. As
consequences of the “$\epsilon-\epsilon^\beta$” property, we derive
some results about the existence and regularity of isoperimetric sets.
This is a joint work with Aldo Pratelli.