Calculus of Variations and Geometric Measure Theory

A. Lemenant

About the regularity of average distance minimizers in R^2

created by lemenant on 29 Jun 2009
modified on 10 Feb 2015


Published Paper

Inserted: 29 jun 2009
Last Updated: 10 feb 2015

Journal: Journal of convex analysis
Year: 2011


We focus on the irrigation problem in R2 and we seek for some conditions to find in the minimizing set S some pieces of C1(or more) regular curves. We prove that it is the case in the ball B when S contains no corner points in B. More generally we prove that the Left and Right tangents half lines of S (that exist everywhere out of endpoints and triple points) are semicontinuous. We also discuss how the regularity is linked with the pull back measure by the projection on S (named PSI). In particular S is $C^{1,alpha}$ in B when PSI is regular with respect to H1 and its density belongs to a certain Lp. We also prove that S is locally a Lipschitz graph away from triple points and endpoints, and that the mean curvature of S is a measure that is related to the measure PSI.