Vol. 254, No. 2, 2011

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The Cheeger constant of curved strips

David Krejčiřík and Aldo Pratelli

Vol. 254 (2011), No. 2, 309–333

We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighborhoods of curves in the plane. If the reference curve is complete and finite (a “curved annulus”), then the strip itself is a Cheeger set and the Cheeger constant equals the inverse of the half-width of the strip. The latter holds true for unbounded strips as well, but there is no Cheeger set. Finally, for strips about noncomplete finite curves, we derive lower and upper bounds to the Cheeger set, which become sharp for infinite curves. The paper is concluded by numerical results for circular sectors.

Cheeger sets, Cheeger constant, curved strips
Mathematical Subject Classification 2010
Primary: 28A75, 49Q20, 35P15, 51M16
Received: 15 November 2010
Revised: 2 October 2011
Accepted: 17 October 2011
Published: 27 February 2012
David Krejčiřík
Department of Theoretical Physics
Nuclear Physics Institute ASCR
25068 Řež
Czech Republic
Aldo Pratelli
Dipartimento di Matematica
Università di Pavia
via Ferrata, 1
I-27100 Pavia