About the Regularity of Average Distance Minimizers in R^(2)

We focus on the following irrigation problem introduced by G. Buttazzo, E. Oudet and E. Stepanov [Optimal transportation problems with free Dirichlet regions, in: Variational Methods for Discontinuous Structures, Progr. Nonlinear Differential Equations Appl. 51, Birkh\"auser, Basel (2002) 41--65]: $$\min \mathcal{F}(\Sigma):=\int_{\Omega}dist(x,\Sigma)\dd \mu(x), \notag$$ where $\Omega$ is an open subset of $\R^2$, $\mu$ is a probability measure and where the minimum is taken over all the sets $\Sigma \subset \Omega$ such that $\Sigma$ is compact, connected, and $\Hh^{1}(\Sigma)\leq \alpha_0$ for a given positive constant $\alpha_0$. In this paper we seek for some conditions to find in $\Sigma$ some pieces of $C^1$ (or more) regular curves. We prove that it is the case in the ball $B$ when $\Sigma \cap B$ contains no corner points. More generally we prove that the Left and Right tangents half lines of $\Sigma$ (that exist everywhere out of endpoints and triple points) are semicontinuous. We also discuss how the regularity is linked with the pull back measure $\psi:= k \sharp \mu$ where $k$ is the projection on $\Sigma$. In particular $\Sigma \cap B$ is $C^{1,\alpha}$ when $\psi$ is regular with respect to $\Hh^1$ with density in a certain $L^p$. We also prove that $\Sigma$ is locally a Lipschitz graph away from triple points and endpoints, and that the mean curvature of $\Sigma$ is a measure that is explicited in terms of measure $\psi$