Published Paper

Inserted: 29 jun 2009
Last Updated: 10 feb 2015

Journal: Journal of convex analysis
Year: 2011

Abstract:

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 L^p. 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.

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• lemenant_regularity2D.pdf