About the Regularity of Average Distance Minimizers in R-2

作者：Lemenant Antoine^*

来源：Journal of Convex Analysis, 2011, 18(4): 949-981.

We focus on the following irrigation problem introduced in [4]
min F(Sigma) := integral(Omega)dist(x, Sigma) d mu(x),
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 of Omega such that Sigma is compact, connected, and H-1(Sigma) <= alpha(o) for a given positive constant alpha(o). 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 boolean AND 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#mu where k is the projection on Sigma. In particular Sigma boolean AND B is C-1,C-o: when psi is regular with respect to H-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.

• 出版日期2011