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The Cheeger constant of curved strips. (English) Zbl 1247.28003

Let \(\Omega\) be an open connected set of the plane \({\mathbb R}^2\). The Cheeger constant is defined as \[ h(\Omega):= \inf_{S\subseteq\Omega}P(S)/|S| \] where the infimum is taken over all sets \(S\subseteq\Omega\) of finite perimeter; \(P(S)\) denotes the perimeter of \(S\) and \(|S|\) the area of \(S\). Any minimizer of the equation above, if it exists, is called a Cheeger set.
By a curved strip one understands a non-self-intersecting tubular neighborhood of a connected \(C^2\)-curve in \({\mathbb R}^2\). There exist infinite, semi-infinite, and finite curved strips as well as curved annuli. The authors prove:
1. In case of a curved annulus or an infinite or a semi-infinite strip the Cheeger constant is the inverse of the half-width of the strip. If the strip is a curved annulus, then the infimum above is attained and the unique Cheeger set coincides with the strip itself. If the strip is infinite or semi-infinite, then there is no Cheeger set.
2. In case of a finite strip, there exists a Cheeger set which is not the whole strip and the Cheeger constant is strictly bigger than the inverse of the half-width of the strip.
Finally, the authors apply their results to the solvable models: annuli, rectangles, sectors.

MSC:

28A75 Length, area, volume, other geometric measure theory
49Q20 Variational problems in a geometric measure-theoretic setting
35P15 Estimates of eigenvalues in context of PDEs
51M16 Inequalities and extremum problems in real or complex geometry
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