Preprint
Inserted: 22 feb 2008
Year: 2008
Abstract:
We construct some examples of explicit solutions to the problem $$ \min\gamma \int\Omega d\gamma(x)\,dx $$ where the minimum is over all connected compact sets $\gamma\subset\overline{\Omega}\subset R^2$ of prescribed one-dimensional Hausdorff measure. More precisely we show that, if $\gamma$ is a $C^{1,1}$ curve of length $l$ with curvature bounded by $1/R$, $l\leq \pi R$ and $\varepsilon\leq R$, then $\gamma$ is a solution to the above problem with $\Omega$ being the $\varepsilon$-neighbourhood of $\gamma$. In particular, $C^{1,1}$ regularity is optimal for this problem.
Download: