On the Cheeger problem for rotationally invariant domains

@article{Bobkov2020OnTC,
  title={On the Cheeger problem for rotationally invariant domains},
  author={Vladimir Bobkov and Enea Parini},
  journal={manuscripta mathematica},
  year={2020}
}
We investigate the properties of the Cheeger sets of rotationally invariant, bounded domains $\Omega \subset \mathbb{R}^n$. For a rotationally invariant Cheeger set $C$, the free boundary $\partial C \cap \Omega$ consists of pieces of Delaunay surfaces, which are rotationally invariant surfaces of constant mean curvature. We show that if $\Omega$ is convex, then the free boundary of $C$ consists only of pieces of spheres and nodoids. This result remains valid for nonconvex domains when the… 

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References

SHOWING 1-10 OF 35 REFERENCES
The Cheeger constant of a Jordan domain without necks
We show that the maximal Cheeger set of a Jordan domain $$\Omega $$Ξ© without necks is the union of all balls of radius $$r = h(\Omega )^{-1}$$r=h(Ξ©)-1 contained in $$\Omega $$Ξ©. Here, $$h(\Omega…
On the higher Cheeger problem
TLDR
The existence of minimizers satisfying additional "adjustment" conditions and study their properties are proved and the results are applied to determine the second Cheeger constant of some planar domains.
On the Minimization of Total Mean Curvature
In this paper we are interested in possible extensions of an inequality due to Minkowski: $$\int _{\partial \Omega } H\,dA \ge \sqrt{4\pi A(\partial \Omega )}$$βˆ«βˆ‚Ξ©HdAβ‰₯4Ο€A(βˆ‚Ξ©) from convex smooth sets…
On the Cheeger sets in strips and non-convex domains
In this paper we consider the Cheeger problem for non-convex domains, with a particular interest in the case of planar strips, which have been extensively studied in recent years. Our main results…
Generalized Cheeger sets related to landslides
Abstract.We study the maximization problem, among all subsets X of a given domain $\Omega$, of the quotient of the integral in X of a given function f by the integral on the boundary of X of another…
The Cheeger constant of curved strips
We study the Cheeger constant and Cheeger set for domains obtained as strip-like neighbourhoods of curves in the plane. If the reference curve is complete and finite (a β€œcurved annulus”), then the…
Isoperimetric estimates for the first eigenvalue of the $p$-Laplace operator and the Cheeger constant
First we recall a Faber-Krahn type inequality and an estimate forp() in terms of the so-called Cheeger constant. Then we prove that the eigenvaluep() converges to the Cheeger constant h() as p β†’ 1.…
CHARACTERIZATION OF CHEEGER SETS FOR CONVEX SUBSETS OF THE PLANE
Given a planar convex domain Q, its Cheeger set CΞ© is defined as the unique minimizer of |βˆ‚X|/|(X| among all nonempty open and simply connected subsets X of Ξ©. We prove an interesting geometric…
Surfaces of revolution with prescribed mean curvature
In this paper we study a surface of revolution in the Euclidean three space Ξ›. The generating curve of the surface satisfies a nonlinear differential equation which describes the mean curvature. The…
A characterization of convex calibrable sets in
Abstract.The main purpose of this paper is to characterize the calibrability of bounded convex sets in by the mean curvature of its boundary, extending the known analogous result in dimension 2. As a…
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