Abstract
We prove the validity of the
1 Introduction
For any
where for every set of locally finite perimeter E we denote by
The isoperimetric problem with single density is a wide generalisation of the classical Euclidean isoperimetric problem, and it has been deeply studied in the last decades, we refer the interested reader to [8, 11, 10, 16, 22, 25, 28, 29, 32] and the references therein. The case of double density is yet a further important generalisation, since many of the possible applications correspond to two different densities. The simplest example is given by Riemannian manifolds, which locally behave as
The isoperimetric problem consists in finding, if they exist, the sets E of minimal perimeter among those of fixed volume. The three main questions one is interested in are existence, boundedness and regularity of isoperimetric sets. Existence in the single density case has been studied in several papers, some of which are those quoted above; for the case of double density it has been studied either for some rather specific choices of the weights (radial [2, 12, 18, 17], monomial [1, 4, 5, 3], Gauss-like [9, 25] or in some Carnot groups [19, 21]), or under very general conditions on f and h, see [20, 30, 33]. Concerning boundedness and regularity, a quite wide answer has been given in the paper [14] for the single density case. The purpose of the present paper is to generalise the results of that paper to the case of double density.
A fundamental tool to study the isoperimetric problem is the classical “
To extend the study to the case of less regular densities, it is convenient to weaken the
Definition 1 (The ε - ε β Property).
Let E be a set of locally finite perimeter and
In the case of single density, in [14] it was shown that whenever f is Hölder continuous with some exponent
Theorem A (The ε - ε β Property).
Assume that f and h are locally bounded, that h is locally α-Hölder in the spatial variable for some
If
Notice that the
Theorem B (Boundedness).
Assume that there exists a constant
and that
Notice that this result, paired with Theorem A, ensures the boundedness of isoperimetric sets in a wide generality, i.e. whenever f and h are bounded and are away from 0 – that is, (1.3) holds – and h is continuous in the spatial variable.
Theorem C (Regularity of isoperimetric sets).
Assume that f and h are locally bounded. Then any isoperimetric set is porous (see Definition 1), and its reduced boundary coincides
This regularity result is not sharp. In particular, in the 2-dimensional case a higher regularity is shown in [13] for the case of single density. Such an improved regularity in the case of a double density and as well in the case
Notice that, since f and h are l.s.c. and positive, they are always locally away from zero. Thus, in Theorems A and C, f and h are locally bounded and locally away from zero, while in Theorem B these requests are made globally. These assumptions are essentially sharp. They can be slightly relaxed to the case of “essentially bounded” or “essentially α-Hölder” functions as defined in [14, Definitions 1.6 and 1.7]. Loosely speaking, this relaxation allows the densities to take the values 0 and
The plan of the paper is as follows. In Section 1.1 we give a short sketch of the proof of Theorem A, to give an idea of how the construction works and to explain where does the constant β in (1.2) come from. In Section 1.2 we recall some basic definitions and properties of sets of finite perimeter. In Section 2 we shall prove Theorem A, which we exploit in Section 3 to prove Theorem B and Theorem C.
1.1 A Quick Sketch of the Proof of Theorem A
Among the three main theorems of this paper, the first one is by far the hardest, and its proof is quite technical. Nevertheless, the overall idea is quite simple, and we outline it in this short section. In particular, the meaning of the value of β in (1.2) will appear as natural. Let us consider a very simplified situation, namely, we assume that the set E is smooth. Of course, the main difficulty of the real proof is exactly to make everything precise also when dealing with non-smooth parts of the boundary.
Let x be a point of
Let us then evaluate the difference between the perimeters of E and F. The boundary of F coincides with the boundary of E except for a “vertical part” (the two vertical segments in the figure) and for the fact that a “horizontal” piece of the boundary has been translated. The extra part consists of two segments of length
Optimising the choice of a and δ subject to the constraint (1.5), we select
from which the above estimate becomes precisely
1.2 Some Properties of Sets of Locally Finite Perimeter
In this short section we recall some basic properties of sets of finite perimeter. A complete reference for the subject is for instance the book [6], however the few things listed below make the present paper self-contained. In this section, whenever we write perimeter we mean the Euclidean one, while by volume the standard N-dimensional Lebesgue measure.
We say that a Borel set
one has
We call the direction
In particular, one defines the perimeter of any set by
Another useful characterisation of the boundary is the following. We say that a set
and we call
Let us now conclude this section by listing two fundamental, well-known results about sets of locally finite perimeter.
Theorem 2 (Blow-Up).
Let
Definition 3 (Vertical and Horizontal Sections).
Given a Borel set
The following theorem, which goes by the name of Vol’pert Theorem, states that almost all (with respect to the proper dimensional Hausdorff measure) vertical sections and horizontal sections of sets of locally finite perimeter are of finite perimeter. Moreover, the reduced boundary of the sections coincides with the sections of the reduced boundary. The proof for vertical sections can be found in [6, 35], while for horizontal ones in [7, 23, 24].
Theorem 4 (Vol’pert).
Let E be a set of locally finite perimeter. Then for
To fully understand the meaning of the Vol’pert Theorem, it is useful to consider what follows. The set
2 Proof of the ε - ε β Property
This whole section is devoted to the proof of the
Proof of Theorem A.
Let
Let
Step I: Choice of the reference cube and the “good” part G. First of all, we let
In fact, the Blow-Up Theorem ensures the first three properties for every small a, and the validity of the last one for some arbitrarily small value of a is then clear by integration. Notice that a depends only on ρ and on E, so ultimately
We want to prove that the set G covers most of the cube Q, and actually only very few perimeter is carried by sections which are not in G. More precisely, we will prove that
To obtain these estimates, let us consider
and by (2.3) we deduce
Instead, Vol’pert Theorem 4 gives that
Since the projection on the first
Therefore, on the one hand by (2.8) and (2.7) we get
On the other hand, (2.1) and (2.2) give
so we deduce
Since
Step II: An estimate about small cubes. In this step we prove a simple estimate about the perimeter on small cubes. More precisely, for every
To prove this estimate, let us fix a direction
We have then
The same estimate is clearly valid replacing
Step III: Selection of “good” horizontal cubes
For each cube, we can find a point
which by (2.11) gives
Keeping in mind that the cubes
provided that
Notice that
Step IV: Choice of one of the horizontal cubes
being G the set defined in Step I. Let us start noticing that everything is trivial for the special case
We focus then on the non-trivial case
hence by (2.12) for more than
The argument to obtain (2.15) is slightly more involved. More precisely, calling A the projection over Q of
It is immediate to notice that ζ is a positive measure and, observing that
which implies (2.15). Summarising, we can find at least one index j with
Step V: Definition of
and we take K pairwise disjoint open strips
Then there exists at least one strip
For the sake of notation, we write
On the other hand, by (2.16) we can estimate
where the last equality follows from Vol’pert Theorem. In particular, for almost every
Consequently, we can find another section
For every
It is easy but important to observe how the set
Step VI: Volume evaluation and choice of the competitor F. In this step we estimate the volume of the sets
We shall then show that such a set is the one required by the
For a given
so that
The last term is the easiest to estimate. Indeed, since of course
Let us now pass to estimate
(2.25)
Let us now instead take
where for every set
Therefore, also by (2.19) we deduce
We are finally in a position to define
Keep in mind that our construction makes sense only if
where the last inequality is true up to taking ρ small enough depending on M and N (keep in mind that ρ was chosen precisely depending on M and on N). On the other hand, in the case
again up to choosing ρ small enough. Since of course the measure of
Step VII: Evaluation of
(2.31)
We also split
(2.32)
We now have to carefully consider the above pieces. The easiest thing to notice is that, since by definition of
so neither E nor F carry any perimeter on the bottom face of the cylinder
Let us now consider
where
To consider the set
where τ is defined in (2.26). As a consequence, also recalling (2.13) we have
To conclude, we have to consider
Instead, by construction,
Putting together (2.33), (2.35), (2.36) and (2.37), and recalling that
for two constants
Step VIII: Optimal choice of γ and definition of β. Estimate (2.38) proved in the above step holds for a generic γ. Keeping in mind that γ could be chosen depending only on N and α, and that the construction requires
With this choice, estimate (2.38) becomes
Step IX: The case of a continuous function h. In this step we consider in more details the case
Let us assume that
Correspondingly, we also modify the definition of
As already done immediately after (2.28), we recall that the construction only makes sense with
as soon as ρ is small enough and depending only on N and M. Again by continuity, we have then the existence of a constant
Exactly in Step VII, we divide
where
To conclude, we need to evaluate the perimeter contribution of
by using the local boundedness, rather we use the continuity of h in the spatial variable. This implies uniform continuity when the spatial variable is inside the cube
Then, calling ξ, as in Step VII, the vertical translation of height δ, and recalling that
Putting this estimate inside (2.37), we now get
where as usual
We can finally conclude. Indeed, the first thing to do is to fix L, depending on
Recalling the definition (2.41) of
Step X: Conclusion (i.e. the case
We can apply this result to the set
Defining then
In other words, we automatically have the validity of (1.1) also for negative values of ε, up to possibly decreasing the value of
3 Boundedness and Regularity of Isoperimetric Sets
In this last section we give the proofs of Theorem B and of Theorem C, which respectively deal with the boundedness and the regularity of isoperimetric sets.
3.1 Boundedness
Let us start with the proof of the boundedness of isoperimetric sets. We underline that the assumptions of the theorem, namely, the boundedness of the densities and the continuity of h, are both necessary. Indeed, even in the special case of single density, it is possible to find unbounded isoperimetric sets when either the boundedness or the continuity assumption is dropped (see [29, 14]).
Proof of Theorem B.
Let E be an isoperimetric set as in the claim. In particular, we can find a ball B and some
Let us fix
which is a bounded and locally Lipschitz decreasing function, thus is particular in
recalling that
holds for
By (1.3) and by the Euclidean isoperimetric inequality we have
Pairing it with the previous estimate, we get
Since
3.2 Regularity
To present the proof of the regularity of isoperimetric sets, we first need to recall some classical definitions and results.
Definition 1.
We say that a set
We say it is ω-minimal, for some continuous and increasing function
We say that a set
Putting together well-known results, see [15, 26, 27, 34], we have the following.
Theorem 2.
Let E be a set of locally finite perimeter. If it is locally quasi-minimal, then it is porous and
We are now ready to prove Theorem C.
Proof of Theorem C.
Let
We fix a generic ball
Let us start to consider the quasi-minimality. First of all, by Theorem A we know that the
Let now
and that
Hence, (3.1) holds true for
If otherwise
so that (3.1) again holds true for
Let us then assume that h only depends on the spatial variable and that it is α-Hölder, so that by Theorem A we have the validity of the
Calling then
By the first part of the proof E is quasi-minimal, thus
where the last inequality comes from the fact that
Funding statement: Both authors are members of the INdAM institute and have been partly supported by the INdAM–GNAMPA 2019 project “Problemi isoperimetrici in spazi Euclidei e non” (project number U-UFMBAZ-2019-000473 11-03-2019).
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