## 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

### Keywords:

Cheeger sets; Cheeger constant; curved strip
Full Text: