Abstract

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, Birkhäuser, Basel (2002) 41–65]: min⁡F(Σ):=∫Ωdist(x,Σ)  dμ(x),\min \mathcal{F}(\Sigma):=\int_{\Omega}dist(x,\Sigma)\dd \mu(x), where Ω\Omega is an open subset of R2\R^2, μ\mu is a probability measure and where the minimum is taken over all the sets Σ⊂Ω\Sigma \subset \Omega such that Σ\Sigma is compact, connected, and H1(Σ)≤α0\Hh^{1}(\Sigma)\leq \alpha_0 for a given positive constant α0\alpha_0. In this paper we seek for some conditions to find in Σ\Sigma some pieces of C1C^1 (or more) regular curves. We prove that it is the case in the ball BB when Σ∩B\Sigma \cap B contains no corner points. More generally we prove that the Left and Right tangents half lines of Σ\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 ψ:=k♯μ\psi:= k \sharp \mu where kk is the projection on Σ\Sigma. In particular Σ∩B\Sigma \cap B is C1,αC^{1,\alpha} when ψ\psi is regular with respect to H1\Hh^1 with density in a certain LpL^p. We also prove that Σ\Sigma is locally a Lipschitz graph away from triple points and endpoints, and that the mean curvature of Σ\Sigma is a measure that is explicited in terms of measure ψ\psi

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Antoine Lemenant

Université Paris Diderot - Paris 7, U.F.R de Mathématiques, Site Chevaleret Case 7012, 175, rue du Chevaleret, 75205 Paris Cedex 13, France

lemenant@ann.jussieu.fr

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Suggested citation

A. Lemenant. “About the Regularity of Average Distance Minimizers in ℝ^2.” Journal of Convex Analysis 18 (2011), No. 4, 949–981. https://doi.org/10.68381/jca18059

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Published by Heldermann Verlag, 2011. Rights now held by Banach Press LLC.