A typical problem in Calculus of Variations is the so-called Cheeger problem: a variational problem in which we seek for the minimum of the ratio P (E) L d (E) among all sets of finite perimeter contained into a given bounded ambient space Ω. Here d stands for the ambient space dimension, P (·) stands for the distributional perimeter (see Subsection 2.3) and L d (E) stands for the Lebesgue measure of E. Formally it can be stated as the infimum problem
.
This problem has been introduced by Jeff Cheeger in [13] with the aim of bounding from below the first eigen-value of the Dirichlet Laplacian on general domains. Lately this problem acquired an interest on its own and the literature produced concerning it is extremely rich, (some contributions can be found at [4] , [5] , [6] , [9] , [10] , [11] , [28] , [29] , [38] ). We also refer to [27] , [32] : two exhaustive surveys on the Cheeger problem in R d . The constant h(Ω) is called the Cheeger constant of Ω and a set E ⊂ Ω attaining P (E) /L d (E) = h(Ω) is called a Cheeger set of Ω. The constant h(Ω) can be interpreted also as the first eigen-value of the p-Laplacian for p = 1, and it can be used to give an upper bound to the diameter of riemannian manifolds bearing nonnegative Ricci curvature (see for instance [26] , [12] ). Existence of Cheeger sets can be attained by direct methods of Calculus of Variations as well as established regularity techniques give further properties of a Cheeger set E in dependence of the regularity of the ambient space Ω (we refer to Theorem 2.5 where the main properties are collected). For instance a strong regularity property of the free boundary ∂E ∩ Ω can be inferred: it is an analytic hyper-surface with constant mean curvature up to a set Σ of dimension at most d − 8. However, an analysis that seems to be missing in literature concerns the size of what we call contact surface of the Cheeger set, consisting of the points where the two boundaries meet ∂E ∩ ∂Ω. Given enough Figure 1 . An ambient space Ω with a representation of one of its Cheeger sets. Provided Ω has boundary regularity of class C 1,1 we can ensure H d−1 (∂E ∩ ∂Ω) > 0. However there can be part of Ω in which, locally, the contact set has smaller dimension than (d − 1). Consider for example the small inner deformation of Ω intoΩ producing a contact set locally finite.
regularity on ∂Ω, a suitable regularity property on ∂E can be derived around contact points, but for example not much is known about how small the set ∂E ∩ ∂Ω can be. We underline that the nature of the problem would suggest that any Cheeger sets will try to be as big as it can (since the ratio P (rE) /L d (rE) = P (E) /rL d (E) scales as 1 /r). So intuitively it appears clear that the contact set ∂E ∩ ∂Ω will be not empty. This intuitive argument however can be pushed up to deduce that H 0 (∂E ∩∂Ω) ≥ 2 as underlined by Remark 2.4 (denoting by H s the s-dimensional measure and for s = 0 the counting measure) but not more. Indeed, for example in presence of high singularities of ∂Ω, such as cusps or angles, the Cheeger set may find convenient to avoid it and prefer a gain in the minimality of the perimeter to a gain in volume, making so unclear what kind of size for ∂E ∩ ∂Ω one should expect. The aim of this work is precisely to fill this lack of knowledge. The answer we are able to provide is extremely precise and it links the regularity of ∂Ω to the Hausdorff dimension of such a set. In particular we are able to prove the following facts, stated detailedly inTheorems 3.1, 3.2, and briefly introduced here.
Theorem. If ∂Ω has regularity of class C 1,α for some α ∈ (0, 1] then
for any E ⊂ Ω Cheeger set. If ∂Ω has regularity of class C 1 then it holds
for any E ⊂ Ω Cheeger set. Moreover for d = 2 these assertions are sharp in the following sense: a) There exists an open bounded set Ω ⊂ R 2 with Lipschitz boundary and having a Cheeger set E ⊂ Ω such that H 0 (∂E ∩ ∂Ω) < +∞; b) for every α ∈ (0, 1] there exists an open bounded set Ω ⊂ R 2 with boundary of class C 1,α such that dim H (∂E ∩ ∂Ω) = α. We make use of the notion of Hausdorff dimension, briefly recalled in Subsection 2.2, in order to describe the size of the contact set. The classification of the contact sets achievable from Theorem 3.1, 3.2 and additional corollaries is reported in Table 1 . It is an easy Corollary then that a regularity of class C 1,1 ensures that H d−1 (∂E ∩ ∂Ω) > 0. We will show also that a brief application of our argument, combined with the well established regularity of Cheeger sets for convex domains, will yield H d−1 (∂E ∩ ∂Ω) > 0 also for Cheeger sets of a convex ambient space Ω.
Let us clarify that there cannot be a local argument that produces a proof for these bounds on the Hausdorff dimension of the contact set. Indeed locally there can be parts of ∂Ω having a local contact surface, with one of its Cheeger sets, of smaller size than the one claimed. Consider for instance the situation in Figure 1 in which we start from a nice bounded set Ω, where H 1 (∂Ω ∩ ∂E) > 0, and we deform locally the part ∂Ω \ ∂E so to obtain someΩ ⊂ Ω such that ∂Ω touches ∂E. Since by moving inward the set Ω until it touches ∂E we do not change its Cheeger constant (and thus neither its Cheeger sets), this construction can easily produce a set of contact points which can have a behavior (locally) as wild as we want. In fact our results states that the global dimension is not going below d − 2 + α but does not exclude that situations as in Figure 1 may happen. This indicates that any argument required to prove our main results 3.1, 3.2 and additional corollaries, has to rely on deductions that take into account the global behavior of the Cheeger set related to ∂Ω.
The argument developed here is indeed a global one which does not produce information on the local behavior of ∂E ∩ ∂Ω. Let us briefly mention the key steps of the argument. We start with an open set Ω with C 1,α boundary and we assume by contradiction that H d−2+α (∂E ∩ ∂Ω) = 0. We know, from general facts about Cheeger sets, that the inner free boundary ∂E ∩ Ω is an analytic hyper-surface with constant mean curvature equal to h(Ω), up to a singular set Σ of Hausdorff dimension at most d − 8. Now the idea is to extend this property to the whole ∂E. If we could do that by exploiting the quantitative smallness assumption on ∂E ∩ ∂Ω, we would be left with a set of finite perimeter and with constant mean curvature, yielding that E must be a ball due to Alexandrov's Theorem [1] . We actually make use of a revisited version of Alexandrov's Theorem, recently proved in [20] and holding for stationary points of the perimeter functional. Now two possibilities are left: either Ω itself was a ball from the very beginning or not. But if Ω was a ball then it is well-known that H d−1 (∂Ω ∩ ∂E) > 0 and this contradicts our assumption H d−2+α (∂E ∩∂Ω) = 0. Thus Ω cannot be a ball and now the contradiction comes from the fact that E would be a ball and a Cheeger set of something which is not a ball. In particular, this contradictions lead us to say that our initial claim H d−2+α (∂E ∩∂Ω) = 0 was false: somewhere it must hold H d−2+α (∂E ∩∂Ω∩A) > 0.
A crucial step in the argument consists thence in extending the constant mean curvature property of E, in force inside Ω (up to a small closed set Σ), to the whole ∂E. Technically, we have a set E with constant distributional mean curvature on R d \ (Γ ∪ Σ), Γ = ∂E ∩ ∂Ω together with some information on the smallness of Γ, Σ and we would like to combine them in order to say that E has constant (distributional) mean curvature on the full R d . The main problems rely in removing Γ.
We make use of the concept of removable singularities, in dealing with the contact set Γ = ∂E ∩ ∂Ω. Let us concentrate on the P DE definition of removability that, consistently to our approach in which we will look at ∂E locally as the graph of a function f E : R d−1 → R, will take place in dimension d = d − 1. Given an open set A ⊂ R d and a differential operator L, a closed set γ ⊂ A is called L-removable if, whenever u is a solution of Lu = 0 weakly in A \ γ then Lu = 0 weakly in A.
The literature on this topic is extremely wide, we recall that some of the pioneering works on this topic are represented by the work of Serrin [39] , and later in [40] for more general elliptic equations in divergence form arising from bounded conservation laws, in which only the gradient of the dependent variable appears explicitly; while on the particular case where L is the nonlinear operator linked to the minimal surface equation, by and Simon [41] . More recent advances in this topic are represented also by the work [19] . Typical statements in this sense infer that if γ is a closed set with null 1-capacity (still in R d ) then γ it is removable. For instance a condition of the type H d −1 (γ) = 0 is enough to ensure that γ is removable for the minimal surface equation. Observe that when translating these results into the context of sets of finite perimeter (and thus in which, by looking at ∂E locally as the graph of a function
does not imply on its own a null 1-capacity of the underlying set γ in R d . Thus we need to investigate on when these kinds of bigger sets can be removed. Concerning this fact we observe that by asking to a weak solution Lu = 0 on A \ γ to be also a slightly more regular u ∈ C 1,α (A) it is possible to remove sets γ with H d −1+α (γ) = 0. This fact had been observed by Pokrovskii in [35] for the minimal surfaces equation, in [33] for the p−Laplacian equation and lately in [34] for uniformly elliptic operators in divergence form. See [36] for an exhaustive survey on removable sets for elliptic operators in the C 1,α class. To our aim, as we need to remove a singularity about a constant mean curvature equation, it would be sufficient to adapt the proof of Pokrovskii in [35] to our case, but as a byproduct of our analysis we observe here that this trade-off in between the regularity of the function and the Hausdorff dimension ensuring the removability of the set γ is actually a sole property of the divergence operator, that can be performed on a general Hölder continuous vector field. Namely, given a vector field F ∈ C 0,α (A) such that
This fact concerning the divergence operator was established by Ponce in [37] , in the case α = 0 and with H d −1 (γ) < +∞, and gives a deep hint on why we can invoke this kind of trade-off in dealing with elliptic equations in divergence form. We state and prove this fact for α > 0 in Section 4 (Proposition 4.2) and then we use it to deduce an alternative proof of the removability of closed sets in the constant mean curvature equation for C 1,α solutions since we think this approach may have an interest on its own. We summarize this in Theorem 4.1 in the context of finite perimeter sets. In particular we have that if u ∈ C 1,α (A) solves −div ∇u
This will be applied to u = f E the local graph representation of ∂E in a neighbourhood of x ∈ ∂E ∩ ∂Ω. In order to complete the extension of the constant mean curvature property on ∂E ∩ ∂Ω by means of these tools, under the assumption H d−2+α (∂E ∩ ∂Ω) = 0, we need just to ensure that ∂E can be locally represented as the graph of a C 1,α function. This is done in Lemma 5.1 in the spirit of [23] by observing that the differential inequality behind the Cheeger problem (Subsection 2.5) allows us to import regularity techniques deeply employed in treating obstacle problems. In particular we obtain that, if ∂Ω itself has C 1,α boundary then ∂E has C 1,α boundary in a neighbourhood of contact points x ∈ ∂E ∩ ∂Ω. Notice that for α = 0 this fact was already observed in [31] in a very general context and for α = 1 in [11] . Up to the best of our knowledge the case α ∈ (0, 1) for Cheeger sets has not been fully developed in literature. However, it was already observed in several treatments of the obstacle problem, mainly for the Dirichlet energy (see for instance [7] ), that the solutions to obstacle problems share the same regularity of the obstacle up to C 1,1 . It is also a commonly observed fact that the regularity obtained in the case of the Dirichlet energy can be transferred to the case of the area functional. Nevertheless it seems that, concerning the Cheeger problem, this argument was never clarified in literature and for the sake of completeness we thus prove it here, starting from a result of [23] through a path involving variational inequalities.
This completes the removal of ∂E ∩ ∂Ω and leads us to infer that E has constant distributional mean curvature in R d \ Σ where Σ has Hausdorff dimension at most d − 8. The set Σ is now easily removable due to its smallness and to the nice structure of E.
We underline that this machinery is actually sensible to the regularity of ∂E more than to the regularity of ∂Ω. In particular if E ⊂ Ω is a Cheeger set with C 1,α boundary then the bounds in the main Theorem still holds true, but concerning just the Cheeger set E.
The examples provided in Figure 1 and the way the argument is developed makes clear that the bound H d−2+α (∂E ∩ ∂Ω) > 0 is really a global bound in the sense that we can have open sets A for which dim H (∂E ∩ ∂Ω ∩ A) < d − 2 + α nonetheless the smallest dimension achievable globally will not go below d − 2 + α. What also arises from the argument is that, if H d−2+α (∂E ∩ ∂Ω ∩ A) = 0 and E has C 1,α boundary in A then ∂E ∩ A must be an analytic hyper-surface with constant mean curvature equal to h(Ω). So the underline boundary of a Cheeger set around contact point that have a local behavior as the one depicted in Figure 1 must be a Constant mean curvature hyper-surface. For this reason we believe that, when Ω is convex, the local bounds on the contact surface with its unique Cheeger set should be true at any scale in the sense that, for any open set A ⊂ R d it should hold that either ∂E ∩∂Ω∩A = ∅ or H d−1 (∂E ∩∂Ω∩A) > 0.
Let us spend also few words on the examples built to prove the sharpness of the bounds in the C 1,α case provided in Section 6. The ingredient required to prove Theorem 3.2 relies on particular solutions of the constant mean curvature equation in dimension one (see Figure 2 and 7). These solutions have the property of solving the ODE everywhere but on a closed set γ which will have the prescribed Hausdorff dimension α. We then suitably sew together these solutions to obtain a set E which is self-Cheeger and that has constant mean curvature equal to H up to a set Γ of Hausdorff dimension α. A suitable family {Ω δ } δ>0 of ambient space Ω is then built around E in a way that ∂Ω δ ∩ ∂E = Γ. The self-Cheeger property and the construction of E, ensures now that, for some δ > 0 there exists an Ω δ that has E as one of its Cheeger sets. As it can be seen we are able to produce examples certifying the sharpness of the dimensional bounds on ∂E ∩ ∂Ω only in dimension d = 2. Notice that, also for d ≥ 3, solutions to the constant mean curvature equation that fails on sets of prescribed Hausdorff dimension appear to exists but the fact that they also have the geometrical properties required to be sewed together, as it can be shown in d = 2, is actually not clear. What appears also to lack are the instruments concerning the self-Cheeger sets (Theorems 6.1,6.3) required to ensures that the profile obtained is self-Cheeger and it remains a Cheeger set of a small perturbation of the ambient space. We believe however, due to the existence of solutions of the constant mean curvature equation failing on sets with prescribed Hausdorff dimension= d − 2 + α, that similar examples can be provided in generic dimension, thus yielding the sharpness of the bounds provided here also for d ≥ 3. We remark also that the ingredient required to produce the example in the Lipschitz case do exploit only the geometrical properties of self-Cheeger sets provided in dimension d = 2 by Theorems 6.1, 6.3 and thus, provided similar instruments are available also in d ≥ 3 it is reasonable to suspect that also for the Lipschitz case the examples can be built in general ambient space dimension. Given to the jump of the dimension of the contact surface, from dim H = 0 in the Lipschitz case to dim H ≥ d − 2 in the C 1 case, it is also reasonable to suspect that, a Lipschitz example for all possible dimensional values between [0, d − 2] of a contact surface can be reached with an explicit construction. We formally state these assertions in the following conjecture. Finally let us note that the approach producing the Hausdorff bounds for contact surface of Cheeger set (and in particular the tools for the removal of the set Γ in Section 4 combined with Theorem 2.4) gives also some information on the structure of the boundary of C 1,α sets with almost-constant mean curvature (See for instance Remark 5.1). In particular by calling
is an analytic hyper-surface with constant mean curvature equal to H
we can see that, if E is a finite perimeter set with C 1,α boundary and
then E is the finite union of balls of radius d−1 H .
We conclude the introduction with a brief description of the organization of the paper. In Section 2 the required background to treat the topics here presented is introduced. In Section 3 the main statements are presented: Theorem 3.1 giving the bounds on the Hausdorff dimension of contact set and Theorem 3.2 inferring the sharpness of these bounds in dimension d = 2. This Section collects also in Table 1 a brief summary of the behavior of the contact surface in dependence of the regularity of ∂Ω arising from our analysis. In Section 4 the analysis of removable singularities for the Divergence operator is carried out. Lately in Section 5 the proof of the dimensional bounds is given. Finally, in Section 6, 2-d Cheeger sets having exactly H α (∂E ∩ ∂Ω) < +∞ are built for all α ∈ (0, 1).
Acknowledgements. The work of MC has been supported by the grant "PRIN 2017 Variational methods for stationary and evolution problems with singularities and interfaces". MC is a member of INdAM-GNAMPA and has been partially supported by the INdAM-GNAMPA Project 2020 "Problemi isoperimetrici con anisotropie" (n.prot. U-UFMBAZ-2020-000798 15-04-2020). The work of SC is partially founded by INdAM (GNAMPA). The authors thank professor Francesco Maggi for the fruitful discussions concerning Proposition 5.1 and its link with Alexandrov's Theorem revisited 2.4. The authors would also like to acknowledge Prof. Giuseppe Buttazzo for the precious discussions concerning the examples in Section 6. Finally the authors are grateful to professor Vincenzo Vespri for his precious feedback and his observations on the overall manuscript. The work in this paper has been almost entirely developed during the COVID-19 lockdown that started in Italy in February 2020.
We here briefly recall the notation and the concepts that will be used in the development of our analysis.
2.1. General notation. In the sequel Ω will denote an open bounded set of R d whose regularity will be specified from time to time. B r (x) will be the ball of radius r centered at x. The same notation will be adopted for cubes Q r (x) by meaning that is a cube of edge length r and centered at x. When r = 1, x = 0 we will simply write B or Q. We denote S d−1 the (d − 1)-dimensional sphere in R d and by e 1 , . . . , e d the canonical basis of R d . For ν ∈ S d−1 we define
By L d we denote the d-dimensional Lebesgue measure and we set ω d := L d (B). We will usually denote by 1 A the characteristic function of a set A:
We will adopt the notation C ⊂⊂ A by meaning that C ⊆ A.
Since the notion of regularity plays a central role in this paper, we set up what we mean for a set A to have boundary regularity of class C k,α locally and globally. To deal with the local graph representation, required to state and manage the definition of regularity, we introduce the following notation. We denote by Q ν r the cube of ν ⊥ with edge length r (centered at 0) and we denote by D ν r (R) = Q ν r × (−R, R)ν the parallelepiped oriented in the direction ν and with a base Q ν r . Often we will assume ν = e d and we will omit to specify it in the notation. We will also do not specify that Q (d−1) r ⊂ R d−1 and we simply write Q r since it will be clear from the context when we are referring to (d − 1)-dimensional cubes.
Definition 2.1 (Boundary regularity for sets in R d ). Let A be a Borel set in R d . We say that ∂A has regularity of class C k,α (for k ∈ N, α ∈ [0, 1]) in a neighbourhood of a point x ∈ ∂A if there exists ν ∈ S d−1 , r, R > 0 and a function f A :
We call (f A , D ν r (R)) the graph representation of ∂A at x (we do not specify the dependence of f A (y) = f x A (y) on the point x to enlighten the notation).
Given a Borel set A ⊂ R d and an open set U ⊂ R d we say that A has boundary of class C k,α in U (and we write ∂A ∈ C k,α (U )) if it has boundary of class C k,α in a neighbourhood of any point x ∈ ∂A ∩ U .
Finally we say that a Borel set A ⊂ R d has boundary of class C k,α (and we simply write ∂A ∈ C k,α ) if it has boundary of class C k,α in a neighbourhood of any x ∈ ∂A.
The same definition applies in the case we refer to a set A with Lipschitz boundary, by meaning that the graph representation invoked has Lipschitz regularity.
In general we say that M is a C k hyper-surface (respectively analytic) if in a neighbourhood of any point y ∈ M there exists a graph representation (f M , D ν r (R)) such that
being ω s a chosen constant. It can be shown that H s δ is an outer measure on R d and it is decreasing in δ (see [30] ). Thus the s-dimensional Hausdorff outer measure H s is well defined as
To briefly clarify this notion let us recall that for any open set A ⊂ R d it holds dim H (A) = d. Moreover we underline, for the sake of clarity, that we will consider the constants ω s to be such that
Remark 2.1. We recall that this notion of dimension is consistent with the usual notion of dimension arising from the geometric setting. Indeed, if M is a k-dimensional manifold in the usual sense it can be shown that dim
is (up to an eventual multiplicative constant) the volume measure on the surface. In particular H k M is locally finite and thus is a Radon measure on R d .
We will require the following characterization, whose proof is postponed to the appendix. We will denote by Q j the countable family of cubes of edge 2 −j yielding the dyadic division of R d into a grid. 2.3. Sets of finite perimeter. We refer the reader to [3] , [30] for a more detailed treatment of this topic.
For a generic Borel set E ⊂ R d and Ω an open set we define the distributional perimeter of E in Ω as
We say that E is a set of finite perimeter if
Thanks to the De Giorgi structure Theorem [14] , [15] , for any set of finite perimeter E, there exists a
∩ Ω) and, for a set E with enough boundary regularity (i.e. regularity of class C 1 ) ∂ * E = ∂E. We say that a set E has distributional mean curvature if there exists a (locally summable)
defined H n−1 -a.e. on ∂ * E. We will sometimes makes use also of the tangential gradient 
Remark 2.2 (On the curvature of the ball). For a set of finite perimeter E with enough boundary regularity (say C 2 regularity), the mean curvature H E : ∂E → R defined above coincides with the usual notion of mean curvature for (d − 1)-dimensional hyper-surfaces, defined as the average of the principal curvatures, up to the multiplicative constant (d − 1). To clarify this point we can see that, for a ball of radius R we can test the above definition with
Lately we will require a sort of decomposition of E in connected components. However the notion of connectedness is not quite appropriate for sets of finite perimeter since a set E can be connected but still be (from the perimeter point of view) the union of several sets which behaves as disjoint with respect to the Perimeter measure. For example if E = B 1 ∪ B 2 is the union of two balls tangent at a point p, this is connected but still P (E) = P (B 1 ) + P (B 2 ). Up to the light of this fact we introduce a more appropriate notion in this context: the notion of indecomposability. Definition 2.3 (Indecomposability for sets of finite perimeter). We say that a set of finite perimeter E is decomposable if there exists two sets of finite perimeter
The following Proposition is a refined version of the well-known Alexandrov's Theorem stated for sets of finite perimeter and it will be extremely useful in our analysis. It comes as a consequence of the analysis developed in [20] , more precisely of results [20, Theorem 1, Corollary 2].
. Let E be a set of finite perimeter and finite volume. Suppose that there exists a constant H > 0 such that
Then E is the union of a finite number of essentially disjoint balls of radius d−1 H . Remark 2.3. Up to the light of Theorem 2.4 and with respect to the Definition of indecomposability 2.3 it is easy to see that an indecomposable set E with constant distributional mean curvature it has to be a single ball.
2.3.1. Constant mean-curvature sets. Let U be an open set and consider u ∈ H 1 (U ) solving, for some H ∈ R −div ∇u
Then it is a well-known fact (see for instance [16] or Theorem 3.2 of [17] or ) that u ∈ C ∞ (U ), analytic and −div ∇u(x)
In particular if a set of finite perimeter E satisfies
we can conclude that ∂ * E ∩ U is an analytic hyper-surface with constant mean-curvature equal to H.
Given an open bounded set Ω the Cheeger constant of Ω (see for instance [27] ) is defined to be 
If Ω is convex then there exists a unique Cheeger set E. Moreover ∂E has regularity of class C 1,1 in a neighbourhood of any x ∈ ∂E ∩ ∂Ω;
Our interest now concerns in study the properties of ∂ * E ∩ ∂ * Ω and in particular its size in dependence on various regularity assumption on Ω. We are interested in determining how small such set can be depending on the regularity of Ω.
Remark 2.4. Notice that, given a general set Ω, it is always possible to ensure that for any Cheeger set E ⊂ Ω of Ω we have H 0 (∂E ∩ ∂Ω) ≥ 2. Indeed, if H 0 (∂E ∩ ∂Ω) = 0 then dist(∂E, ∂Ω) > 0 and we can dilate a bit E into λE ⊂ Ω, λ > 1 in order to obtain P (λE)
contradicting the fact that E was a Cheeger set. But also, if H 0 (∂E ∩ ∂Ω) = 1 we can find a small translation τ such that E + τ ⊂ Ω and with dist(τ + ∂E, ∂Ω) > 0. Then we could apply again the dilation argument and violate the minimality property of E. So H 0 (∂E ∩ ∂Ω) ≥ 2.
In the proof of Theorem 3.1 we will require E to be indecomposable. Let us briefly clarify this point.
Remark 2.5. Let E ⊂ Ω and suppose that E has two indecomposable components (the general case follows easily by arguing in the exact same way), say E 1 , E 2 . Then we claim that
In particular this shows that if E is a Cheeger set then one of its indecomposable component is also a Cheeger set of Ω and this justifies why, without loss of generality, in the proof of Theorem 3.1 we can move from E to one of its indecomposable component. Indeed if it were the case
In particular
and thus
Definition 2.6 (self-Cheeger sets). We say that a set of finite perimeter E is a self-Cheeger set if it holds
Cheeger problem as an obstacle problem. The relation between the graph representation of E and the graph representation of Ω will be repeatedly recalled in the sequel. Let Ω be an open bounded set with ∂Ω ∈ C k,α for some k ≥ 1, α ∈ (0, 1). Let E ⊂ Ω be one of its Cheeger sets. Then recall that Assertion (II) of Theorem 2.5 states that the closed set (∂E \ ∂ * E) ∩ Ω has Hausdorff dimension at most d − 8 and that at any
r ) analytic and solving the constant mean curvature equation
Moreover (Assertion (III)-(IV)-(V) of Theorem 2.5) at any point x ∈ ∂E ∩ ∂Ω, ν E (x) = ν Ω (x) and we can find a graph representation
for all x ∈ γ r,R .
Moreover f E provides also a solution to the area minimizing obstacle problem with prescribed mean-curvature
2.6. Campanato spaces and regularity theory tools. Given a smooth bounded domain A ⊂ R d , a function u ∈ L 2 (A) is said to belong to the Campanato space L 2,λ (A) if there exists C,˜ > 0 such that for each x ∈ A and ≤˜ there holds
Here and in the sequel
We recall that the celebrated results due to Campanato [8] , infer the existence of an isomorphism between L 2,λ and C 0,α for α = λ−d 2
1
.
An important tool that we will use crucially is the following technical Lemma for whose proof we refer to the book [2] .
then holds the estimate φ( ) ≤ C a for all ≤ * for a constant C = C(A, B, τ, a, b, d, * , σ) but independent of φ.
In this Section we present our results and their consequences on the size of the contact surface.
We present our result concerning the contact surface of Cheeger sets in terms of the regularity of ∂Ω. As it can be viewed by the argument run to prove Theorem 3.1, the result depends really only on the regularity of ∂E. A regularity argument for the obstacle problem allows us to infer that any Cheeger set shares at least the same regularity of the ambient space Ω and this permits to derive Theorem 3.1 in terms of the regularity of ∂Ω. We decided to present the Theorem in this way for clarity of exposition but let us stress that the important ingredient is represented by regularity of ∂E and that Theorem 3.1 remains true if we replace the hypothesis on ∂Ω with a C 1,α regularity hypothesis on the boundary of one of its Cheeger set ∂E. Clearly then the bounds hold just for that Cheeger set. and Ω ⊂ R d be an open bounded set with ∂Ω ∈ C 1,α . Then any E Cheeger set of Ω has boundary regularity of class C 1,α in a neighbourhood of any x ∈ ∂E ∩ ∂Ω and
As a consequence
Furthermore, if α = 0 and Ω is an open bounded set with ∂Ω ∈ C 1 then for any E Cheeger set of Ω we have additionally
As a partial proof of the sharpness of the dimensional bounds given by Theorem 3.1 we are able to provide examples, in dimension d = 2, saturating the estimates (3.1), (3.2) . Furthermore there is an open bounded set with Lipschitz boundary Ω having a Cheeger set E ⊂ Ω satisfying H 0 (∂E ∩ ∂Ω) < +∞. In particular this really tells us that, if we hope to ensure that ∂E ∩ ∂Ω achieve a set of full H d−1 measure we are forced to ask that Ω has at least C 1,1 -boundary (or that it is convex, see Corollary 3.2). For any α ∈ (0, 1) there exists an open bounded set Ω ⊂ R 2 with boundary regularity of class C 1,α having a Cheeger set E ⊂ Ω such that 
We summarize the classification of the dimension of the contact set arising from Remark 2.4 Theorems 3.1, 3.2, and Corollaries 3.1, 3.2 in Table 1 .
The main topic of this Section is the understanding of the removability of sets for distributional mean curvature. In particular a proof of the following Theorem is given. 
Then ∂E ∩ U is an analytic hyper-surface with constant mean curvature equal to H.
As already explained in the introduction, Theorem 3.1 can be derived by invoking the results in [36] . We present here an alternative proof, based on Proposition 4.2: a slight extension to the case of α > 0 of the work of Ponce [37] , developed in much more generality in the case α = 0 and restated below suited to our case in Proposition 4.1, since we think it may have an interest on its own and can lead to possible futher extensions of the analysis carried on in [36] .
Remark 4.1. The existence of the sets stated in Theorem 3.2 for α ∈ (0, 1) provides also a partial proof of the sharpness for Theorem 4.1 in d = 2. Indeed the Cheeger set E of Ω produced will have ∂E ∈ C 1,α and it will solve H E = h(Ω) on R d \ Γ where Γ = ∂E ∩ ∂Ω satisfies 0 < H α (Γ) < +∞. In particular ∂E is not an analytic profile.
There exists an open bounded set Ω with Lipschitz boundary and E ⊂ Ω Cheeger set such that
For any α ∈ (0, 1) there exists an open bounded set Ω with boundary of class C 1,α and E ⊂ Ω Cheeger set
No example known Table 1 . The behavior of the contact surface in dependence of the regularity of ∂Ω.
Theorem 4.1 gives us a precise criterion that tells us when we can extend the constant distributional mean curvature locally, a priori constant everywhere but undefined on a set Γ whose smallness depends on the regularity of ∂E.
The following Propositions are stated in dimension d but, since they will be applied to the graph representation in order to prove Theorem 4.1, they will be invoked in dimension d = d − 1. 
We extend Proposition 4.1 to the case α > 0 in the next Proposition. Proof. Let ϕ ∈ C ∞ c (A) and call K = spt(ϕ), γ = K ∩ γ which is compact and satisfies H d−1+α (γ ) = 0. We fix ε > 0 and we invoke Proposition 2.1 to find j ∈ N such that Q j := {Q j k (x j k )}, the countable family of closed cubes centered at {x j k } k∈N and of edge length 2 −j tasselling R d , satisfies
Since γ is compact we can find a finite number of cubes (relabelled here for the sake of shortness) from the grid
Call
where here, and from now on, C stands for a constant independent of j, ε and that may vary from line to line. Moreover we have ˆ∂
Notice also that we havê
where we do not explicitize the dependence of G on the index . Therefore
Notice that the above estimate holds trivially also for α = 0 with a constant in place of 2 −jα . In particular ˆ∂
We thus can conclude that ˆA F · ∇ϕdx −ˆA gϕdx ≤ Cε, and since C is independent of ε, j and the above is in force for any fixed ε > 0 we can derivê A F · ∇ϕdx =ˆA gϕdx.
The above can be repeated for any ϕ ∈ C ∞ c (A 
Then u ∈ C ∞ (A) and
−div ∇u(x) 
Then u ∈ C ∞ (A) and
−div ∇u(x)
Proof of Theorem 4.1. We show only the case α > 0, since the case α = 0 follows in the exact same way provided we apply Corollary 4.1 in place of Corollary 4.2.
We first observe that, since ∂E ∈ C 1,α (U ), then ∂E = ∂ * E on U . For any z ∈ ∂E ∩ U we can represent locally E around z as the graph of a C 1,α function. If z / ∈ Γ then for a small > 0 we have Γ∩(∂E ∩B (z)) = ∅ and (∂E ∩ B (z)) ⊂ ∂E ∩ U . In particular we have that E has constant distributional mean curvature on B (z) and thus ∂E ∩ B (z) is an analytic hyper-surface with constant mean curvature equal to H (see Subsection 2.3.1).
Suppose then that z ∈ (∂E ∩ Γ) ∩ U (assume z = 0, ν E (0) = ν Ω (0) = e d , ∇f E (0) = ∇f Ω (0) = 0) and consider Q r ⊂ R d−1 , (f E , D r (R) = Q r × (−R, R)) the graph representation of ∂E around z = 0, for some r, R > 0, f E ∈ C 1,α (Q r ) (see Subsection 2.5, ν = e d has been omitted for the sake of shortness). Set F (y) := (y, f E (y)) ∈ C 1,α (Q r ; D r (R)), call Γ r,R := (∂E ∩ Γ) ∩ D r (R) and observe that Γ r,R := F (γ r,R ) where γ r,R := {y ∈ Q r | F (y) ∈ Γ r,R }. Notice that
and thus (see Subsection 2.3.1) f E ∈ C ∞ (Q r \ γ r,R ) and it solves also
Moreover the function F is a Lipschitz function and up to suitably decrease r, R we can infer that also
We thus have Q r ⊂ R d−1 , f E ∈ C 1,α (Q r ) ∩ C ∞ (Q r \ γ r,R ), H (d−1)−1+α (γ r,R ) = 0 and f E solves strongly the constant mean curvature equation outside of the closed set γ r,R . We are exactly in the hypothesis of Corollary 4.2 and thus we can conclude f E ∈ C ∞ (Q r ) and
In particular f E is analytic. Thence also for z ∈ (∂E ∩ Γ) ∩ U , for small > 0, ∂E ∩ B (z) is an analytic hyper-surface with constant mean curvature equal to H. Therefore ∂E ∩ U is an analytic hyper-surface with constant mean curvature equal to H. Let Ω be an open bounded set with ∂Ω ∈ C 1,α , for some α ∈ [0, 1] and let E ⊂ Ω be one of its Cheeger sets. Then ∂E has boundary regularity of class C 1,α in a neighbourhood of any x ∈ ∂Ω ∩ ∂E.
Proof. The extremal cases α = 0, 1 are stated in Assertions (IV) and (V) of Theorem 2.5. Let α ∈ (0, 1).
We fix x ∈ ∂E ∩ ∂Ω and we invoke assertion (IV) of Theorem 2.5 to deduce that ∂E has C 1 regularity in a neighbourhood of x and that ν E (x) = ν Ω (x) = ν. Assume without loss of generality that x = 0, ν = e d and let f Ω , f E , D r (R) = Q r × (−R, R) (where D r (R) = D ν r (R), and ν = e d has been omitted for the sake of shortness), Q r ⊂ R d−1 , r, R > 0 the graph representation of ∂Ω, ∂E respectively f Ω , f E : Q r → (−R, R). Then the regularity of ∂Ω, ∂E at x = 0 yields f Ω ∈ C 1,α (Q r ), f E ∈ C 1 (Q r ). We can also assume without loss of generality that
We observe further that it will be sufficient to conclude the desired regularity in a neighbourhood of x ∈ ∂E ∩ ∂Ω that f E ∈ C 1,α (Qr /2 ); and to this aim, we will demonstrate that for there is a * < r /2 such that for all x ∈ Qr /2 and for all 0 < 1 ≤ 2 ≤ * , β > α, and for A, B > 0 independent of the chosen point x ∈ Qr /2 ,
Indeed the above, combined with the fact that
will allow us, by invoking Lemma 2.
to infer that φ x ( ) ≤ C 2α+d for all ≤ * As in the thesis of Lemma 2.1 the constant C depends on A, B, α, β, d, * , σ but not on φ x , provided A, B in (5.1) do not depend on x ∈ Qr /2 , we have that
uniformly in x ∈ Qr /2 , ≤ * . This yields ∇f E ∈ L 2,2α+d (Qr /2 ) which, combined with the Campanato isomorphism recalled in Subsection 2.6, gives f E ∈ C 1,α (Qr /2 ). We now focus in proving (5.1) by dividing the proof in two steps, for the sake of conciseness.
Step one: Reduction to a variational inequality. We consider the obstacle problem (2.5) to which f E is a solution, to show how this leads to a variational inequality. Let us define the convex set As for each u ∈ K, t ∈ [0, 1] we have that f E − t(f E − u)) ∈ K and thus I(f E ) ≤ I(f E − t(f E − u)). In particular, this yields d dt t=0
and thus the variational inequality
Step two: The comparison technique. To begin, we consider 0 < 1 < 2 ≤ r /2, and we split the function
). By classic results on regularity for solutions to quasilinear equations as (5.3) we know that U ∈ C ∞ (Q 2 (x)) and in particular that for all ε > 0, 1 ≤ 2 , we have the estimate
for c = c(d) and that in the sequel may vary from line to line. Let us also denote with f x, := f Q (x) to ease the notation. We estimate the averaged integral of ∇f E aŝ
Last inequality is due to the following consideration. Let us show directly that
Firstly we observe that, the U function satisfies to the following equation
). Secondly we recall that the map z → z √ 1+|z| 2 satifies the following fine properties :
Monotonicity: there exists ν > 0 such that for all z 1 , z 2 ∈ R n (5.8)
Boundedness: by the Lipschitz character of the map z → z/ √ 1 + z 2 there exists a µ > 0 such that
Thus we introduce ϕ(y) = (U (y) − (∇U ) x, 2 · y) + ((∇U ) x, 2 · y − f E (y)) in equation (5.7) to obtain, by the properties (5.8), (5.9) and the use of Young inequality ab ≤ εa p + C(ε)b p , a, b > 0 when p, p are conjugate exponents,
Hence by choosing ε = (2c) −1 we can reabsorb the smaller term on the left hand side of the inequality to get
(∇f E − ∇U ) dx 2 dy implying (5.6) by an application of Jensen's inequality to the last term.
To accomplish the Campanato estimate (5.1) on f E , we estimate the last quantity on the right hand of the inequality of (5.5). We get back to (5.2) writing
h Ω (f E − v)dy , thus by using monotonicity and boundedness of mean curvature operator and Young's inequality as before we infer
· ∇ϕ dy, (5.11) so that, by inserting in (5.11) ϕ = U − v and with the help of Young's inequality again, we obtain
Hence by letting ε = (2c) −1 and exploiting the regularity of the obstacle f Ω ∈ C 1,α (Q r ) we get finally
Last term in (5.10) can be estimated with Poincaré and Hölder inequality aŝ
so that for a radius small enough we can reabsorb the first term on the right and obtain the required estimate
Gathering together (5.5), (5.13) we obtain (5.1), as desired.
Removability of small sets in the interior. We will require, in order to deal with the negligible set Σ produced by Assertion (II) of Theorem 2.5, of the following Proposition which can be derived from Theorem 2.4.
Proposition 5.1. Let E be a set of finite perimeter and Σ ⊂ ∂E be a closed set such that H d−2 (Σ) = 0.
Suppose that E, Σ have the following properties: a) There exists C 0 , r 0 depending on Σ such that
Then E is a finite union of balls of radius d−1 H .
Notice that Proposition 5.1 cannot be deduced by invoking Theorem 4.1 since no information on the regularity of ∂E is given on Σ but property a). In some sense this is consistent with the De Giorgi-Stampacchia-Simon Theorem for minimal surface equation [18] , [41] , since H d−2 (Σ) = 0, combined with the flatness mild regularity property a) around point x ∈ Σ, can be interpreted as an information on the 1-capacity (in the d = (d − 1)-dimensional space) of Σ.
Proof. Let C 0 , r 0 be the constants given by property a). By invoking Proposition 2.1, for any ε > 0 we can find a j ∈ N and a finite number of cubes Q j (x 1 ), . . . , Q j (x kj ) ∈ Q j of edge length j = 2 −j such that
Up to futher increase j we can also infer that j ≤ r0 8d . Subordinated to this proof we introduce the short notation
Then ζ is piece-wise smooth and for almost every x ∈ R d satisfies (see for instance [42] ) Pick now δ << j so small that
Notice that ζ δ ∈ C ∞ (R d ) and
Moreover, by omitting the center of the cubes,
Henceforth, since also
Pick any
In particular, by invoking the property of r 0 , for all i = 1, . . . , k j we have
for a constant C uniform for x ∈ Σ. Thus we obtain
Analogously we can obtain
and the above can be repeated for all T ∈ C ∞ c (R d ; R d ). In particular we are in the position for apply Theorem 2.4 and conclude that E must be a finite union of balls of radius d−1 H . A crucial ingredient in order to apply Proposition 5.1 is thence the following density estimate for constrained perimeter minimizers. This argument is extremely common in problem of these kinds. For the sake of self-containment of the paper we have added a brief proof.
Let Σ ⊂ ∂E be a closed set. Then there exists a constant C 0 = C 0 (Σ) > 0 and r 0 = r 0 (Σ) > 0 such that
Proof. We recall the following volume-fixing variation argument from [30, Lemma 17.21] . If E is a set of finite perimeter and A is an open set such that H d−1 (∂ * E ∩ A) > 0, then there exist σ 0 = σ 0 (E, A) > 0 and C = C(E, A) < ∞ such that for every σ ∈ (−σ 0 , σ 0 ) we can find a set of finite perimeter F with F ∆E ⊂⊂ A and
) be the constants given by the volume-fixing variation Lemma.
We set r 0 < min
Then we can find a set of finite perimeter F with F ∆E ⊂⊂ B 0 (x) and such that
We now define F := (E \ B r (y)) ∩ B r (y) ∪ F ∩ B r (y) c and we notice that, since F = E on B r (y), by exploiting (5.20) we can achieve
This, combined with the fact that E was a Cheeger set and that by construction F ⊆ Ω, gives
Moreover, since P F ; B r (y) = P E \ B r (y); B r (y) ≤ dω d r d−1 we have also 
and since (5.24) is in force for all y ∈ Σ and for all r < r 0 = r 0 (Σ) we conclude.
Proof. We start with the Proof of Theorem 3.1 and then we briefly explain how to adapt it to obtain Corollary 3.2.
Proof of Theorem 3.1. We provide only the proof for the case α > 0 since the case α = 0 again will follow in the exact same way.
Let Ω be an open set with ∂Ω ∈ C 1,α . We immediately invoke Lemma 5.1 to deduce that ∂E has boundary regularity of class C 1,α in a neighbourhood of any point
Assertion (II) of Theorem 2.5 tells us that ∂ * E ∩ Ω is an analytic hyper-surface of constant mean curvature equal to h(Ω) and that the singular closed set Σ := (∂E \ ∂ * E) ∩ Ω has Hausdorff dimension at most d − 8.
Suppose that H d−2+α (∂E ∩ ∂Ω) = 0. Since Σ ⊂ Ω is closed, and due to the regularity of ∂E close to ∂Ω, we can find two open sets U ∂Ω , U Ω with the following properties
(iii) ∂E ∈ C 1,α (U ∂Ω ); In particular ∂E ∩U ∂Ω = ∂ * E ∩U ∂Ω and ∂ * E ∩U Ω is an analytic hyper-surface with constant mean curvature equal to h(Ω). Setting Γ = ∂E ∩ ∂Ω, E has constant distributional mean-curvature on U ∂Ω \ Γ. Since we assumed that H d−2+α (Γ) = 0 and since ∂E ∈ C 1,α (U ∂Ω ) we invoke Theorem 4.1 to infer that ∂E ∩ U ∂Ω is an analytic hyper-surface with constant mean curvature equal to h(Ω). In particular we have that E has constant mean curvature equal to h(Ω) on (U Ω ∪ U ∂Ω ) \ Σ which, since E ⊂ Ω ⊂⊂ U Ω ∪ U ∂Ω , is equivalent to say that E has constant mean curvature equal to h(Ω) on R d \ Σ. Moreover Σ ⊂ ∂E, H d−2 (Σ) = 0. Thence Property b) of Proposition 5.1 holds. But E is a Cheeger set and (5.25) is in force. Then we can invoke Lemma 5.2 to conclude that property a) of Proposition 5.1 holds as well on E, Σ. This is enough to apply Proposition 5.1 to conclude that E must be a finite union of balls of radius d−1 h(Ω) . Since, because of Remark 2.5, we can move from E to one of its indecomposable component we can reduce to the case where E is a single ball of radius d−1 h(Ω) . But now
we bump into a contradiction. This contradiction is a consequence of the fact that we exploited that H d−2+α (∂E ∩ ∂Ω) = 0 to extend the validity of the constant mean curvature equation also on ∂E ∩ ∂Ω by means of Theorem 4.1. In order not to have the contradiction it must hold, for some x ∈ ∂E ∩ ∂Ω, r > 0, that H d−2+α (B r (x) ∩ (∂E ∩ ∂Ω)) > 0.
Remark 5.1. Notice that the same proof produces also information on the size of the region where E do not has constant mean curvature, relatively to its boundary regularity. In particular, for an indecomposable set of finite perimeter E, by denoting with (5.26) Cmc(∂E; H) := x ∈ ∂E B r (x) ∩ ∂E is an analytic hyper-surface with constant mean curvature equal to H , we have that, whenever ∂E ∈ C 1,α , either E is a ball or
Indeed, if we assume that E is not a ball and we violate (5.27) we can get a contradiction by arguing similarly to the proof of Theorem 3.1 with the aid of Theorems 2.4, 3.1. As a byproduct we can apply this criterion to self-Cheeger sets by concluding that, if E is an indecomposable self-Cheeger set with C 1,α boundary then either E is a ball or
Proof of Corollaries 3.1, 3.2. The proof of Corollary 3.1 is trivial since we just apply Theorem 3.1 to the case α = 1. To obtain Corollary 3.2 we assume H d−1 (∂E ∩ ∂Ω) = 0 and we apply the same argument as in the proof of Theorem 3.2 to derive a contradiction, by invoking Assertion (VI) of Theorem 2.5 in place of Lemma 5.1.
We provide here a geometric construction leading to a set Ω having a Cheeger set E with C 1,α boundary, such that H α (∂E ∩ ∂Ω) < +∞ and dim H (∂E ∩ ∂Ω) = α.
The construction will be done in several step by exploiting Cantor stair-case type functions.
6.1. Technical tools. We invoke the following criterion from [28, Theorem 1.1] suitably re-written according to our necessities.
Theorem 6.1 (Self-Cheeger criterion). If E ⊂ R 2 is a simply connected open bounded set with Lipschitz boundary and at any point x ∈ ∂E there exists a ball of radius r = L 2 (E) P (E) tangent to x and entirely contained in E then E is self-Cheeger, i.e.
We also require the following Definition.
Definition 6.2. Let Ω ⊂ R 2 be a simply connected open set with C 1 boundary. We say that Ω has no neck of radius r if for any x, y ∈ Ω such that dist(x, ∂Ω), dist(y, ∂Ω) > r there exists a continuous C 1 curve γ : [0, 1] → Ω such that B r (γ(t)) ⊆ Ω for all t ∈ [0, 1].
Concerning the above Definition, we recall the following result from [28, Theorem 1.4] , suitably re-written in our framework to fit our construction and in force for the ambient space dimension d = 2. where Ω r := {x ∈ Ω | dist(x, ∂Ω) > r}.
Moreover r = h(Ω) −1 is the unique solution to the equation
A central tool in the analysis in [28] , giving the result 6.3, is the following extension of Steiner formulas to any simply connected domain with Lipschitz boundary and no neck of radius r, that we report here since they will be of some use in the sequel: L 2 (Ω) = L 2 (Ω r ) + M 0 (Ω r )r + πr 2 (6.2) P (Ω) = M 0 (Ω r ) + 2πr (6.3) where M 0 (Ω r ) is the Minkowski content of ∂Ω r which is finite for Ω with no neck of radius r and Lipschitz boundary.
6.1.1. Cantor sets and Cantor staircase properties. We consider the well-known iterative procedure that produces a Cantor-type set. To fix the notation we briefly report it here. We set, for τ ∈ (0, 1), C 1 (τ ) = [0, 1] \ 1−τ 2 , 1+τ 2 and then we define C n (τ ) as the set obtained from C n−1 (τ ) by removing, on each of its connected components, the central interval of length τ (1−τ ) n−1
Notice that each C n−1 (τ ) is made by 2 n−1 disjoint interval and therefore
Setting C(τ ) := +∞ n=1 C n (τ ) to be the limiting Cantor-type set and defined (6.4) α(τ ) = log (2) log 2 1−τ then it is a well-known fact that dim H (C(τ )) = α(τ ) (see for details on generalized Cantor functions [25] or [21] , [22] for general theory of fractals). By varying τ ∈ (0, 1) we can reach all α ∈ (0, 1). Consider
It is a well-known fact that s n,τ uniformly converges on [0, 1] to a Cantor-type staircase function s τ which is C 0,α ([0, 1]) ∩ C ∞ ((0, 1) \ C(τ )) for α identified by (6.4) . We collect here some elementary properties of s τ . More precisely, for fixed H, ∈ R + we consider From now on only values H < 2 will be considered. We state the following Lemma, inferring some properties of u τ , derived from the properties of s τ . We refer to Figure 2 where few iterations are depicted. Lemma 6.2. For any H < 2, τ ∈ (0, 1) the function u τ (·; H, ) satisfies the following properties: Figure 2 . The first iterations u n,τ (defined as the u τ but with s n,τ in place of s τ ) of the function u τ for the case τ = 1/3. In yellow is depicted the Cantor-type set while the blue part is the region of constant mean curvature. The orange part lying above the Cantor-type set is the region where the function fails to solve the constant mean curvature ODE. For the case τ = 1/3 the function u τ is well defined up to H < 6 and, since for small value of H the oscillatory effect is not quite visible, in order to magnify the behavior of u τ the parameter have been set to be H = 5.5, = 1. which implies
Hence, setting
we have P (t) = 0 which gives the desired symmetry on u τ .
In order to prove some additional properties of u τ , required to run the construction, we need also the following Lemma. Lemma 6.3. For any H < 2, τ ∈ (0, 1) it holds that
for all t ∈ [0, ] and for any r ∈ (0, ) ∩ H −1 (s τ (t; H, )) − 1, s τ (t; H, )) + 1).
Proof. We omit to specify H, in the argument of u τ , s τ for the sake of shortness. Notice that the function
is C 1 ((0, )) and
since s τ is a non decreasing function and r < t. Notice that the function z → z √ 1 − z 2 is also non decreasing and thus P t (r) ≤ 0 on r ≤ t.
Analogous computation yields also that P t (r) ≥ 0 on r ≥ t yielding that r = t is a point of global minimum for P and achieving the proof.
We finally show the key Propositions 6.1, 6.2. Proposition 6.1. For any H < 2, τ ∈ (0, 1) the function u τ (·; H, ) satisfies the following property : for any x ∈ {(t, u τ (t; H, )) | t ∈ (0, )} there exists a unique ball B 1/H tangent to x and entirely contained in the epigraph of u τ (·; H, ).
Proof. Once again the parameter H, have been omitted in the proof for the sake of shortness. Pick t ∈ (0, ), let (6.7) q 0 (t) = (t, u τ (t)) + 1
and consider the ball B 1/H (q 0 (t)). Then clearly (t, u τ (t)) ∈ ∂B 1/H (q 0 (t)). We now prove that it lies below the graph of u τ . Call
the upper part of ∂B 1/H (q 0 (t)) which lies in [0, ] × R (see Figure 3 ). For any p ∈ a q0(t) it suffices to prove that p · e 2 ≤ u τ (p · e 1 ) which means
for all r ∈ [0, ] ∩ (q 0 (t) · e 1 − H −1 , q 0 (t) · e 1 + H −1 ). This becomes Figure 3 . In the picture the analysis of the profile u τ carried on in Proposition 6.1 and Figure 4 . A depiction of Lemma 6.4: The yellow region represents the circular sector C, the blue line the arc a and the red line the curve ϑ. On the left is depicted the case in which the circular sector spans an angle less than or equal to π/2 on the right the case in which the circular sector spans an angle bigger than π/2 but strictly less than π. Thus sin β 2 < τ < 1 and hence β/2 < π/2 yielding β < π.
We collect an easy geometrical fact, that will be useful in the proof of Theorem 3.2.
Lemma 6.4. Let C be a circular sector of radius r relative to an arc a spanning an angle β. Call x 0 , y 0 the left and right extremum of the arc respectively. Let ϑ : [0, 1] → R 2 be a curve lying outside of C and such that ϑ(0) = x 0 , ϑ(1) = y 0 . If β ≤ π/2 then for any x ∈ C we have dist(x, ϑ) ≤ r.
If β ∈ (π/2, π) then there exists a value δ 0 = δ 0 (β, r) > 0 such that if δ < δ 0 and dist(a, ϑ) ≤ δ then for any x ∈ C dist(x, ϑ) ≤ r. Proof. We will make use of Figure 4 to help the reader in following the proof. We just observe that, if β ≤ π/2, the union of the two circles centered at x 0 , y 0 and with radius r covers C (see picture 4). Thus at any
If β ∈ (π/2, π) call U := C \ (B r (x 0 ) ∪ B r (y 0 )) = ∅. Let a δ be an arc of radius r + δ and spanning an angle β from the lines on which x 0 , y 0 lies (see Figure 4 ). Then ϑ is forced to lie in between the arc a and the arc a δ . Notice that all balls of radius r centered at a point in a δ are tangent to a circle of radius δ centered at the origin. This means that if δ < δ 0 (r, β) < 2r sin π − β 2 = (β, r)
since (β, r) is the length of the segment connecting the origin to the intersection point between ∂B r (x 0 ) and ∂B r (y 0 ) in ∂U . Let then x ∈ U and let p ∈ a δ be such that x ∈ B r (p). Then, since ϑ is connecting x 0 to y 0 , there is a point z ∈ ϑ such that the segment connecting x to p intersects ϑ in z. But then dist(x, ϑ) ≤ |x − z| ≤ |x − p| ≤ r.
We are now ready to prove Theorem 3.2.
Proof of Theorem 3.2. To show how the construction is performed we start by dealing with the Lipschitz case. The C 1,α case will follow a similar argument, based on Theorems 6.1, 6.3 and Lemma 6.4.
The Lipschitz case. Fix k ∈ N big enough and consider the regular k-gon R k of the plane with edge length . On each edge consider an arc of radius H −1 and spanning an angle smaller than π from the j-th vertex of the k-gon to the (j + 1)-th vertex (we refer to figure 5 with k = 6). Notice that this procedure is possible with an angle β( , H) < π, provided < 2 H , since clearly (6.8) = 2H −1 sin( β /2).
I H := 0, 2 H and for ∈ I H call E( ) such a set. We now show that, for some 0 ∈ I H it holds (6.9)
Indeed the map → L 2 E( ) H −1 is continuous and for small
Moreover, as an easy computation shows, we have that
For → 2/H (6.8) gives β → π and hence (6.10) lim
Notice that 1 tan( π /k) − π 2 ≥ 0 ⇔ k ≥ π arctan( 2 /π) ≈ 5.5.
Thence for k ≥ 6 we can achieve also
This means that, for any k ≥ 6, some 0 ∈ (0, 2 /H) satisfying (6.9) exists. We also notice that E( 0 ) has no neck of radius H −1 . Therefore we invoke Steiner formulas (6.2), (6.3), that, combined with (6.9), yields
) . Now the very definition of E( 0 ) tells us that at any z ∈ ∂E( 0 ) there is a ball of radius H −1 entirely contined in E( 0 ) and tangent at z. This fact, combined with Theorem 6.1 and the equality H −1 = L 2 (E( 0)) P (E( 0 )) , implies thence that E is self-Cheeger and that h(E) = H −1 . Theorem 6.3 tells us now that we can re-write E( 0 ) as union of balls of radius H −1 :
Consider now Ω δ to be a small perturbation of ∂E( 0 ) in a way that ∂Ω δ ∩∂E( 0 ) = {the family of vertexes} and dist(∂Ω δ , ∂E( 0 )) ≤ δ (see Figure 6 ). We now claim that, for some δ small enough, Ω δ has no neck of radius H −1 and (6.11) Figure 6 . The way ∂Ω δ is built once 0 has been calibrated depending on H. Notice that all the angles β < π ensures that we can apply Lemma 6.4 and conclude that, for small δ, E( 0 ) is a Cheeger set of Ω δ as well.
This will imply, thanks again to Theorem 6.3, that E( 0 ) is a Cheeger set of Ω δ . The contact set is now given just by the vertexes and thus H 0 (∂E( 0 ) ∩ ∂Ω δ ) = k with Ω δ an open bounded set with Lipschitz boundary. Since clearly, for small values of δ, Ω δ has no neck of radius H −1 , let us thus focus on (6.11).
Any connected component S of ∂E( 0 ) \ ∂Ω δ is made by an arc a S spanning an angle strictly smaller than π. Call C S the circular sector relative to the arc a S and split
If instead z ∈ E( 0 ) \ E( 0 ) H −1 then either z ∈ T or z ∈ R. If z ∈ T (see again Figure 6 ) then it belongs to some C S relative to an arc a S spanning an angle less than π. In particular by invoking Lemma 6.4 we can find a δ such that for all z ∈ C S it holds
Since there are a finite number of arcs we can find a small δ for which dist(z, ∂Ω δ ) ≤ H −1 holds for every z ∈ T . If instead z ∈ R we simply observe that
Thence (6.11) is in force and the construction of the example is concluded.
The C 1,α case. The logic of the proof is essentially the same of the one in the Lipschitz case. Consider the construction yielding the Cantor-type set C(τ ) and the function u τ . For τ = 1 we have α = α(τ ) = 0 (defined in (6.4)) and for τ = 0 we have α = 1. We will produce examples for α ∈ (0, 1) thus choose τ ∈ (0, 1) and fix H ∈ R + . Then consider where 0 has been chosen suitably so that (6.13) is in force. The darker part of ∂E( 0 ) is the part of the boundary obtained as the graph of u n,τ while the ligther part are the quarter of circles of radius 1/H that has been used to surger together the profiles of u n,τ in the four directions. The grey set in the middle is the set E( 0 ) H −1 defined by (6.12). By calibrating δ we can obtain the sought set Ω δ with the aid of Theorems 6.1, 6.3.
For any ∈ I H consider a set E( ) made by the four copies of (t, u τ (t; H, )), as in Figure 7 , joined by four quarter of circle of radius H −1 . Observe that if we surger the graph of u τ with a quarter of circle as in Figure  7 we preserve the C 1,α regularity of the whole profile, due to the fact that u τ (0; H; ) = u τ ( ; H, ) = 0 and that u τ is C α in (0, ) as stated in Lemma 6.2. Consider
We show, as in the Lipschitz case, that there exists an 0 ∈ I H such that
In this case the estimates can be easily done by observing that E( ) H −1 always contains a square of edge length . Thence, for > √ π H (which is an admissible value in I H since √ π < 2) we have
Moreover notice that any E( ) is contained in a ball of radius √ 2 ) and thus
In particular for < 1 H √ 2 (which is again an admissimble value in I H ) we have
Since the map → L 2 E( ) H −1 is continuous the intermediate value theorem tells us that there is an 0 such that (6.13) is satisfied. We now have that H −1 satisfies (6.13), and E( 0 ) has no neck of radius 1/H (since E( 0 ) H −1 is path connected). By exploiting Steiner formulas (6.2), (6.3) as in the Lipschitz case we can see that H −1 = L d (E( 0)) P (E( 0)) . Thus, by applying Theorem 6.1 and by arguing as in the Lipschitz case (with the aid of Proposition 6.1 in applying Theorem 6.1) we can conclude that E( 0 ) is self-Cheeger and that h(E( 0 )) = H −1 . In particular by applying 6.3 we can see also that
The ambient space Ω δ can be now built by gently pushing up ∂E( 0 ) far away from the Cantor set: i.e. by considering an Ω δ represented by the function f δ (t) := u(t) + δg(t) on [0, ] with g(t) ∈ C ∞ ((0, )) a non-negative smooth function such that C(τ ) = {g = 0}, and a regular small surgery on the part where E( 0 ) is made by quarter of circles (see again Figure 7 ). This function has the same regularity of u, namely C 1,α , which yields that Ω δ has C 1,α boundary regularity. As in the Lipschitz case we just need to prove that, for δ small enough, (6.14)
Ω H −1 δ = E( 0 ) H −1 and then Theorem 6.3 will ensure that E( 0 ) is a Cheeger set of Ω δ as well. But now, by construction ∂E( 0 ) ∩ ∂Ω δ has the same dimension of the underlying Cantor-type set C(τ ) used to build the profile u τ and thus dim H (∂E( 0 ) ∩ ∂Ω δ ) = α. This would complete the construction. Therefore let us focus on proving (6.14).
For x ∈ Ω δ \ E( 0 ) we have, for δ small enough, dist(x, ∂Ω δ ) ≤ δ < H −1
and thus E( 0 ) H −1 ⊂ Ω H−1 δ . We just need to prove that for x ∈ E( 0 ) \ E( 0 ) H −1 we have dist(x, ∂Ω δ ) ≤ H −1 .
Propositions 6.1, 6.2 and the construction of E( 0 ) tell us that on any connected component S of ∂E( 0 )\∂Ω δ is a circular arc a S from a circle of radius 1/H and centered on ∂E( 0 ) H−1 (by definition of E( 0 ) H −1 ) touching ∂E( 0 ) ∩ ∂Ω δ in exactly two points x S 0 , y S 0 and entirely lying below ∂Ω δ elsewhere. Call C S the circular sector identified by a S and notice that we can split E( 0 ) \ E( 0 ) H −1 as Figure 8 ). We consider two cases.
Case one: x ∈ T . In this case we have that x ∈ C S for some S connected component of ∂E( 0 ) \ ∂Ω δ . We now notice, thanks to Proposition 6.2 and to the construction of E( 0 ), that all the circular sectors of T , but the four central ones C S1 , . . . , C S4 , span an angle less or equal than π/2. Moreover the remaining sectors C S1 , . . . , C S4 which can span an angle bigger than π/2 spans an angle β i (τ ) < π still due to Proposition 6.2. In particular we can invoke Lemma 6.4 to conclude that there is a δ 0 = δ 0 (β 1 (τ ), . . . , β 4 (τ ), H) such that if δ < δ 0 dist(x, ∂Ω δ ) ≤ H −1
and thus x / ∈ Ω H −1 δ .
Case two: x ∈ R. In this case we immediately have dist(x, ∂Ω δ ) = dist(x, ∂E( 0 ) ∩ ∂Ω δ ) = dist(x, ∂E( 0 )) ≤ H −1 and then again x / ∈ Ω H −1 δ . Figure 8 . We can split the portion E( 0 ) \ E( 0 ) H −1 into a portion T (in yellow) made by circular sectors C S of radius H −1 , corresponding to the connected componend S of ∂E( 0 )\∂Ω δ , and a portion R (in green) which is the remaining part. In the case x ∈ T we are in the situation of Lemma 6.4 due to the fact that each C S , but the central ones C S1 , . . . , C S4 , do not span an angle bigger than π/2. The central ones instead span an angle strictly less than π and thus for δ small enough we have dist(x, ∂Ω) ≤ H −1 for x ∈ T . If instead x ∈ R we simply observe that |x − y| = dist(x, ∂Ω δ ) = dist(x, ∂E( 0 ) ∩ ∂Ω δ ) = dist(x, ∂E( 0 )) ≤ H −1 .
Remark 6.2. Notice that a careful analysis of the proof of Theorem 3.2 in the Lipschitz case tells us that we can produce, for any k ≥ 6, an open bounded set with Lipschitz boundary and having a Cheeger set E such that H 0 (∂E ∩ ∂Ω) = k. The question of whether the value k = 6 represents some sort of threshold below which we cannot go in performing the construction, arises naturally. In particular, for k ≤ 6 what fails in the argument is the positivity of the term and since → L 2 E( ) H −1 is increasing we would also have
This forbids us from apply Theorems 6.1, 6.3 and we cannot guarantee that E( 0 ) will be self-Cheeger for some values of 0 ∈ (0, 2 /H) (actually it will not be self Cheeger for any value of ∈ (0, 2/H)). It is not the purpose of this analysis to investigate further this question and thus we post-pone the treatment of this topic to future work.
Proof of Proposition 2. It is a well known fact that H s is a measure on R d and is called the dyadic Hausdorff measure. It can also be shown that (see for instance [21] for a full treatment of the whole topic)
for a constant C d,s depending on d, s only. Therefore, if H s (N ) = 0 then H s (N ) = 0. In particular, for any ε > 0 there is a j 0 such that H s ,j (N ) ≤ ε for all j > j 0 . Thus there is k > j > j 0 such that
Cheeger problem as an obstacle problem. Let us briefly treat the obstacle problem of which the graph representation f E is a solution, for E Cheeger set. Assume that ∂Ω ∈ C 1 and let x ∈ ∂E ∩ ∂Ω (assume x = 0, ν E (x) = ν Ω (x) = e d ). Consider the graph representation of E in D r (R) = Q r × (−R, R), Q r ⊂ R d−1 , f E : Q r ⊂ R d−1 → (−R, R), r, R > 0, f E ∈ C 1 (Q r ) (Assertion (III) Theorem 2.5). Then P (E; D r (R)) =ˆQ
Let w ∈ H 1 0 (Q r ), w = f E on ∂Q r and w ≤ f Ω on Q r . We set F w := (E \ D r (R) c ) ∪ {(x, s) | x ∈ Q r , s ≤ w(x)} Notice that we have still F w ⊂ Ω. Moreover E = F w on D r (R) c and P (F w ; D r (R)) =ˆQ r 1 + |∇w| 2 dx, L d (F w ∩ D r (R)) =ˆQ r wdx.
This yields h(Ω) ≤ P (E; D r (R) c ) +´Q r 1 + |∇w| 2 dx L d (E ∩ D r (R) c ) +´Q r wdx which is, by invoking (7.1),
This means that f E solves (2.5). Morever, this minimality property allows us also to conclude that, for all ϕ ∈ C ∞ c (Q r ), ϕ ≥ 0, it holdsˆQ r ∇u · ∇ϕ 1 + |∇u| 2 dx ≤ˆQ h(Ω)ϕdx.
Equivalently, being the above valid for all ϕ ≥ 0 we have −div ∇u 1 + |∇u| 2 ≤ h(Ω) weakly on Q r .