On the Cheeger sets in strips and non-convex domains. (English) Zbl 1337.49074

The Cheeger problem is an isoperimetric-like problem. It has connections to Riemannian geometry, calculus of variations and eigenvalues problem. For a domain \(\Omega \subseteq \mathbb{R}^n\) one looks for an estimate of the Cheeger constant, defined as \(h(\Omega) =\inf{\{{ {P(F)}\over {|F|}}}: F\subseteq \Omega, |F|>0\}\), where \(|F|\) and \(P(F)\) denote, respectively, the volume and the perimeter of a Borel set \(F\subseteq\mathbb{R}^n\). A set \(F\subseteq \Omega\) which realizes the above infimum is called a Cheeger set in \(\Omega\). After stating some basic properties of the Cheeger sets, the authors obtain the continuity of the Cheeger constant. Then, they discuss the Cheeger problem in convex domains obtaining a certain uniqueness and convexity result. They obtain some further results about Cheeger sets in \(\mathbb{R}^2\). Some characterizations of Cheeger sets in planar strips are given. Among these characterizations Theorem 3.2 is notable. Let \(S\) be also an open strip of length \(L\) and width 2. Then, \(h(S)=1+{{\pi}\over{2L}}+O(L^{-2})\) as \(L\to +\infty\).
Then the authors prove some simple technical relations between the perimeter and the area of a strip and the length of the corresponding spinal curve. The final result is Theorem 3.3., asserting the existence and the uniqueness of a Cheeger set for a certain open strip \(S\). Next, the authors provide some planar examples.


49Q10 Optimization of shapes other than minimal surfaces
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35P15 Estimates of eigenvalues in context of PDEs
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