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.

MSC:

 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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References:

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