In this paper we provide an approximationà la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the so-called Steiner Problem, but also the average distance problem, and finally a problem relying on the p-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of Γ-convergence relies on technical tools from geometric measure theory, as for instance a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets.
In this article, we consider and analyse a small variant of a functional originally introduced in [9, 22] to approximate the (geometric) planar Steiner problem. This functional depends on a small parameter ε > 0 and resembles the (scalar) Ginzburg-Landau functional from phase transitions. In a first part, we prove existence and regularity of minimizers for this functional. Then we provide a detailed analysis of their behavior as ε → 0, showing in particular that sublevel sets Hausdorff converge to optimal Steiner sets. Applications to the average distance problem and optimal compliance are also discussed.
Relying on the effect of microscopic asperities, one can mathematically justify that viscous fluids adhere completely on the boundary of an impermeable domain. The rugosity effect accounts asymptotically for the transformation of complete slip boundary conditions on a rough surface in total adherence boundary conditions, as the amplitude of the rugosities vanishes. The decreasing rate (average velocity divided by the amplitude of the rugosities) computed on close flat layers is definitely influenced by the geometry. Recent results prove that this ratio has a uniform upper bound for certain geometries, like periodical and "almost Lipschitz" boundaries. The purpose of this paper is to prove that such a result holds for arbitrary (non-periodical) crystalline boundaries and general (non-smooth) periodical boundaries.
Received (Day Month Year) Revised (Day Month Year) Communicated by (xxxxxxxxxx)We present a quantitative analysis of the effect of rough hydrophobic surfaces on viscous newtonian flows. We use a model introduced by Ybert and coauthors in Ref. 20, in which the rough surface is replaced by a flat plane with alternating small areas of slip and noslip. We investigate the averaged slip generated at the boundary, depending on the ratio between these areas. This problem reduces to the homogenization of a non-local system, involving the Dirichlet to Neumann map of the Stokes operator, in a domain with small holes. Pondering on the works of Allaire (see Ref. 2, 3) we compute accurate scaling laws of the averaged slip for various types of roughness (riblets, patches). Numerical computations complete and confirm the analysis.
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