The Cheeger constant of curved strips

Abstract

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.

A Faber–Krahn Inequality for the Cheeger Constant of $$N$$ N -gons

“…The minimization problem (1), named after Cheeger who introduced it in [11], has attracted a lot of interest in recent years; without any attempt at completeness, a list of related works is [1,2,[8][9][10]14,17,18,22,25,26,29]. Here we limit ourselves to recalling that, for as above, there exists at least a solution to (1), which is called a Cheeger set of , and in general is not unique (unless is convex; see [1]).…”

On the p-torsion functions of an annulus

“…(In the bidimensional case, the calibrability of an annulus can also be obtained as consequence of results presented in [10].) In this paper we prove some properties of the p-torsion functions for the annulus Ω a,b as well as results connecting these functions with the Cheeger constant of Ω a,b .…”

“…In order to prove the upper bound in (15) we make use of the Pohozaev's identity (10). Since the left-hand side of (10) is positive and the right-hand side is…”

Anisotropic mean curvature on facets and relations with capillarity

“…The case of strips has been investigated in [57] in the Euclidean setting. Our aim is to generalize it to the anisotropic setting.…”

“…In order to prove the statement, recalling also Remark 4.8, we want to construct a selection with divergence constantly equal to 1 a . Following [57], (29) we define the vector field̃︀ N on F as︀…”