Optimal Thin Torsion Rods and Cheeger Sets

Abstract

We carry out the asymptotic analysis of the following shape optimization problem: a given volume fraction of elastic material must be distributed in a cylindrical design region of infinitesimal cross section in order to maximize the resistance to a twisting load. We derive a limit rod model written in different equivalent formulations and for which we are able to give necessary and sufficient conditions characterizing optimal configurations. Eventually we show that, for a convex design region and for very small volume fractions, the optimal shape tends to concentrate section by section near the boundary of the Cheeger set of the design. These results were announced in [11].

Perspective Functions: Properties, Constructions, and Examples

“…In the special case when G = R, ϑ arises in mechanics [3,15] as well as in statistics [53], where it is called the Berhu (or reverse Huber) function. The reason for this terminology is that (3.6) exhibits a quadratic behavior on B(0; ρ) and a sublinear behavior outside, while (3.12) exhibits a sublinear behavior on B(0; ρ) and a quadratic behavior outside.…”

A Faber–Krahn Inequality for the Cheeger Constant of $$N$$ N -gons

“…The minimization problem (1), named after Cheeger who introduced it in [11], has attracted a lot of interest in recent years; without any attempt at completeness, a list of related works is [1,2,[8][9][10]14,17,18,22,25,26,29]. Here we limit ourselves to recalling that, for as above, there exists at least a solution to (1), which is called a Cheeger set of , and in general is not unique (unless is convex; see [1]).…”

Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

“…Even more specifically, it is enough to consider the class O of octagons in K 2 (meant as the class of polygons with at most 8 sides, so that it includes hexagons and squares). Restricted to such class, the variant of problems (6)-(7) reads (8) sup…”

“…where Per(E, R 2 ) denotes the perimeter of E in the sense of De Giorgi. In recent years, the minimization problem (11), named after Cheeger who introduced it in [14], has attracted an increasing interest: without any attempt of completeness, see [1,8,11,10,12,13,16,17,18,24,25,33]; further references can be found in the review papers [26,29]. Below we recall the main results that we shall need to exploit about the Cheeger problem.…”