Cheeger Sets Research Articles

The Cheeger problem in abstract measure spaces

AbstractWe consider nonnegative ‐finite measure spaces coupled with a proper functional that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

The Cheeger cut and Cheeger problem in metric measure spaces

In this paper we study the Cheeger cut and Cheeger problem in the general framework of metric measure spaces. A central motivation for developing our results has been the desire to unify the assumptions and methods employed in various specific spaces, such as Riemannian manifolds, Heisenberg groups, graphs, etc. We obtain two characterization of the Cheeger constant: a variational one and another one through the eigenvalue of the 1-Laplacian. We obtain a Cheeger inequality along the lines of the classical one for Riemannian manifolds obtained by Cheeger in (In: Gunning RC (ed) Problems in analysis. Princeton University Press, Princeton, pp 195–199, 1970). We also study the Cheeger problem. Through a variational characterization of the Cheeger sets we prove the existence of Cheeger sets and obtain a characterization of the calibrable sets and a version of the Max Flow Min Cut Theorem.

Overdetermined problems and relative Cheeger sets in unbounded domains

In this paper, we study a partially overdetermined mixed boundary value problem for domains \Omega contained in an unbounded set \mathcal{C} . We introduce the notion of the Cheeger set relative to \mathcal{C} and show that if a domain \Omega\subset\mathcal{C} admits a solution of the overdetermined problem, then it coincides with its relative Cheeger set. We also study the related problem of characterizing constant mean curvature surfaces \Gamma inside \mathcal{C} . In the case when \mathcal{C} is a cylinder, we obtain further results whenever the relative boundary of \Omega or the surface \Gamma is a graph on the base of the cylinder.

The role of Cheeger sets in the steady flows of viscoplastic fluids in pipes: A survey

Given a convex or a Jordan domain Ω, let Ω′ be a subset of this domain, with P(Ω′) denoting its perimeter and A(Ω′) its area. If a subset Ωc exists such that h=P(Ωc)/A(Ωc) is a minimum, the subset Ωc is called the Cheeger set of Ω and h, the Cheeger constant of the given domain. If one considers the reciprocal of this minimum or the maximum ratio of the area of the subset to its perimeter, t∗=1/h. It follows from the work of Mosolov and Miasnikov that the minimum pressure gradient G to sustain the steady flow of a viscoplastic fluid in a pipe, with a cross section defined by Ω, is given by G>τy/t∗, where τy is the constant yield stress of the fluid. In this survey, we summarize several results to determine the constant h when the given domain is self-Cheeger or a Cheeger-regular set that touches each boundary of a convex polygon and when the Cheeger-irregular set does not do so. The determination of the constant h for an arbitrary ellipse, a strip, and a region with no necks is also mentioned.

Cheeger sets and the minimum pressure gradient problem for viscoplastic fluids

Cheeger sets and the minimum pressure gradient problem for viscoplastic fluids

Extensions and traces of BV functions in rough domains and generalized Cheeger sets

Extensions and traces of BV functions in rough domains and generalized Cheeger sets

Dimensional lower bounds for contact surfaces of Cheeger sets

We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Ω⊂Rd. By providing bounds on the Hausdorff dimension of the contact surface ∂E∩∂Ω, we show a fruitful interplay between this size itself and the regularity of the boundaries. Eventually, we obtain sufficient conditions to infer that the contact surface has positive (d−1) dimensional Hausdorff measure. Finally we prove by explicit examples in two dimensions that such bounds are optimal.

Cheeger Sets for Rotationally Symmetric Planar Convex Bodies

In this note we obtain some properties of the Cheeger set C_varOmega associated to a k-rotationally symmetric planar convex body varOmega . More precisely, we prove that C_varOmega is also k-rotationally symmetric and that the boundary of C_varOmega touches all the edges of varOmega .

On the Cheeger problem for rotationally invariant domains

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 generating curve of $C$ is closed, convex, and of class $\mathcal{C}^{1,1}$. Moreover, we provide numerical evidence of the fact that, for general nonconvex domains, pieces of unduloids or cylinders can also appear in the free boundary of $C$.

Minimizers of the prescribed curvature functional in a Jordan domain with no necks

We provide a geometric characterization of the minimal and maximal minimizer of the prescribed curvature functionalP(E) −κ|E| among subsets of a Jordan domainΩwith no necks of radiusκ−1, for values ofκgreater than or equal to the Cheeger constant of Ω. As an application, we describe all minimizers of the isoperimetric profile for volumes greater than the volume of the minimal Cheeger set, relative to a Jordan domainΩwhich has no necks of radiusr, for allr. Finally, we show that for such sets and volumes the isoperimetric profile is convex.

On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum/the max of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity. In the specific case of the max of the first Robin eigenvalue, we are able to remove the convexity assumption on the cells.

Two examples of minimal Cheeger sets in the plane

We construct two minimal Cheeger sets in the Euclidean plane, i.e. unique minimizers of the ratio "perimeter over area" among their own measurable subsets. The first one gives a counterexample to the so-called weak regularity property of Cheeger sets, as its perimeter does not coincide with the $1$-dimensional Hausdorff measure of its topological boundary. The second one is a kind of porous set, whose boundary is not locally a graph at many of its points, yet it is a weakly regular open set admitting a unique (up to vertical translations) non--parametric solution to the prescribed mean curvature equation, in the extremal case corresponding to the capillarity for perfectly wetting fluids in zero gravity.

The Cheeger constant of a Jordan domain without necks

We show that the maximal Cheeger set of a Jordan domain $\Omega$ without necks is the union of all balls of radius $r = h(\Omega)^{-1}$ contained in $\Omega$. Here, $h(\Omega)$ denotes the Cheeger constant of $\Omega$, that is, the infimum of the ratio of perimeter over area among subsets of $\Omega$, and a Cheeger set is a set attaining the infimum. The radius $r$ is shown to be the unique number such that the area of the inner parallel set $\Omega^r$ is equal to $\pi r^2$. The proof of the main theorem requires the combination of several intermediate facts, some of which are of interest in their own right. Examples are given demonstrating the generality of the result as well as the sharpness of our assumptions. In particular, as an application of the main theorem, we illustrate how to effectively approximate the Cheeger constant of the Koch snowflake.

Weighted Cheeger sets are domains of isoperimetry

We consider a generalization of the Cheeger problem in a bounded, open set $\Omega$ by replacing the perimeter functional with a Finsler-type surface energy and the volume with suitable powers of a weighted volume. We show that any connected minimizer $A$ of this weighted Cheeger problem such that $H^{n-1}(A^{(1)} \cap \partial A)=0$ satisfies a relative isoperimetric inequality. If $\Omega$ itself is a connected minimizer such that $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$, then it allows the classical Sobolev and $BV$ embeddings and the classical $BV$ trace theorem. The same result holds for any connected minimizer whenever the weights grant the regularity of perimeter-minimizer sets and $\Omega$ is such that $|\partial \Omega|=0$ and $H^{n-1}(\Omega^{(1)} \cap \partial \Omega)=0$.

Morrey spaces and generalized Cheeger sets

Abstract We maximize the functional ∫ E h ⁢ ( x ) ⁢ 𝑑 x P ⁢ ( E ) , \frac{\int_{E}h(x)\,dx}{P(E)}, where E ⊂ Ω ¯ {E\subset\overline{\Omega}} is a set of finite perimeter, Ω is an open bounded set with Lipschitz boundary and h is nonnegative. Solutions to this problem are called generalized Cheeger sets in Ω. We show that the Morrey spaces L 1 , λ ⁢ ( Ω ) {L^{1,\lambda}(\Omega)} , λ ≥ n - 1 {\lambda\geq n-1} , are natural spaces to study this problem. We prove that if h ∈ L 1 , λ ⁢ ( Ω ) {h\in L^{1,\lambda}(\Omega)} , λ > n - 1 {\lambda>n-1} , then generalized Cheeger sets exist. We also study the embedding of Morrey spaces into L p {L^{p}} spaces. We show that, for any 0 < λ < n {0<\lambda<n} , the Morrey space L 1 , λ ⁢ ( Ω ) {L^{1,\lambda}(\Omega)} is not contained in any L q ⁢ ( Ω ) {L^{q}(\Omega)} , 1 < q < p = n n - λ {1<q<p=\frac{n}{n-\lambda}} . We also show that if h ∈ L 1 , λ ⁢ ( Ω ) {h\in L^{1,\lambda}(\Omega)} , λ > n - 1 {\lambda>n-1} , then the reduced boundary in Ω of a generalized Cheeger set is C 1 , α {C^{1,\alpha}} and the singular set has Hausdorff dimension at most n - 8 {n-8} (empty if n ≤ 7 {n\leq 7} ). For the critical case h ∈ L 1 , n - 1 ⁢ ( Ω ) {h\in L^{1,n-1}(\Omega)} , we demonstrate that this strong regularity fails. We prove that a bounded generalized Cheeger set E in ℝ n {\mathbb{R}^{n}} with h ∈ L 1 ⁢ ( ℝ n ) {h\in L^{1}(\mathbb{R}^{n})} is always pseudoconvex, and any pseudoconvex set is a generalized Cheeger set for some h.

On the generalized Cheeger problem and an application to 2d strips

In this paper we consider the generalization of the Cheeger problem which comes by considering the ratio between the perimeter and a certain power of the volume. This generalization has been already sometimes treated, but some of the main properties were still not studied, and our main aim is to fill this gap. We will show that most of the first important properties of the classical Cheeger problem are still valid, but others fail; more precisely, long and thin rectangles will give a counterexample to the property of Cheeger sets of being the union of all the balls of a certain radius, as well as to the uniqueness. The shape of Cheeger set for rectangles and strips is then studied as well as their Cheeger constant.

Critical Yield Numbers of Rigid Particles Settling in Bingham Fluids and Cheeger Sets

We consider the fluid mechanical problem of identifying the critical yield number $Y_c$ of a dense solid inclusion (particle) settling under gravity within a bounded domain of Bingham fluid, i.e., the critical ratio of yield stress to buoyancy stress that is sufficient to prevent motion. We restrict ourselves to a two-dimensional planar configuration with a single antiplane component of velocity. Thus, both particle and fluid domains are infinite cylinders of fixed cross-section. We then show that such yield numbers arise from an eigenvalue problem for a constrained total variation. We construct particular solutions to this problem by consecutively solving two Cheeger-type set optimization problems. Finally, we present a number of example geometries in which these geometric solutions can be found explicitly and discuss general features of the solutions.

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 are an estimate on the Cheeger constant of strips, which is stronger than the previous one known from Krejciřik and Pratelli (Pac J Math 254(2):309–333, 2011), and the proof that strips share with convex domains a number of crucial properties with respect to the Cheeger problem. Moreover, we present several counterexamples showing that the same properties are not valid for generic non-convex domains.

Planar infinite-horizon optimal control problems with weighted average cost and averaged constraints, applied to cheeger sets

We establish a Poincare-Bendixson type result for a weighted averaged infinite horizon problem in the plane, with and without averaged constraints. For the unconstrained problem, we establish the existence of a periodic optimal solution, and for the constrained problem, we establish the existence of an optimal solution that alternates cyclicly between a finite number of periodic curves, depending on the number of constraints. Applications of these results are presented to the shape optimization problems of the Cheeger set and the generalized Cheeger set, and also to a singular limit of the one-dimensional Cahn-Hilliard equation.

Shape derivative of the Cheeger constant

This paper deals with the existence of the shape derivative of the Cheeger constant $h_1(\Omega)$ of a bounded domain $\Omega$. We prove that if $\Omega$ admits a unique Cheeger set, then the shape derivative of $h_1(\Omega)$ exists, and we provide an explicit formula. A counter-example shows that the shape derivative may not exist without the uniqueness assumption.