In this work, we prove the crossover in the coarsening rates in terms of time-averaged lower bounds on the energy, which scales like an inverse length. We use a method proposed by Kohn and the first author [15], which exploits the gradient flow structure of the dynamics. Our adaption uses techniques from optimal transportation. Our main ingredient is a dissipation inequality. It measures how the optimal transportation distance changes under the effects of convective and diffusive transport.

]]>E(γ) = Z Rd d(x, Γγ) p dµ(x) + λ Length(γ)

where γ : I → R d , I is an interval in R, Γγ = γ(I), and d(x, Γγ) is the distance of x to Γγ. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure H1 , and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures µ supported in two dimensions the minimizing curve is injective if p ≥ 2 or if µ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation.

]]>(0.1) E λ µ (Σ) := Z Rd d(x, Σ)dµ(x) + λH1 (Σ),

among pathwise connected, closed sets, Σ. Here d(x, Σ) is the distance from a point to a set and H1 is the 1-Hausdorff measure. In a sense the problem is to find a onedimensional measure that best approximates µ. It is known that the minimizer Σ is topologically a tree whose branches are rectifiable curves. The branches may not be C 1 , even for measures µ with smooth density. Here we show a result on weak second-order regularity of branches. Namely we show that arc-length-parameterized branches have BV derivatives and provide a priori estimates on the BV norm. The technique we use is to approximate the measure µ, in the weak-∗ topology of measures, by discrete measures. Such approximation is also relevant for numerical computations. We prove the stability of the minimizers in appropriate spaces and also compare the topologies of the minimizers corresponding to the approximations with the minimizer corresponding to µ.

]]>The average-distance problem is to find the best way to approximate (or represent) a given measure *μ * on R^{d}Rd by a one-dimensional object. In the penalized form the problem can be stated as follows: given a finite, compactly supported, positive Borel measure *μ*, minimize E(Σ)=∫Rdd(x,Σ)dμ(x)+λH1(Σ)

among connected closed sets, *Σ *, where λ>0λ>0, d(x,Σ)d(x,Σ) is the distance from *x* to the set *Σ *, and H^{1}H1 is the one-dimensional Hausdorff measure. Here we provide, for anyd⩾2d⩾2, an example of a measure *μ * with smooth density, and convex, compact support, such that the global minimizer of the functional is a rectifiable curve which is not C^{1}C1. We also provide a similar example for the constrained form of the average-distance problem.