The ε - ε^β Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets

Definition 1 (The ε-εβ Property).

Let E be a set of locally finite perimeter and β∈[0,1]. We say that E possesses the ε-εβ property (relative to the densities f and h) if for any ball B such that ℋN-1⁢(B∩∂*⁡E)>0 there exist constants C>0 and ε¯>0 such that for all |ε|<ε¯, there exists a set F such that

Theorem A (The ε-εβ Property).

Assume that f and h are locally bounded, that h is locally α-Hölder in the spatial variable for some α∈[0,1], and that E⊆RN is a set of locally finite perimeter. Then E possesses the ε-εβ property, where β is given by

If α=0 (in which case locally α-Hölder precisely means locally bounded) and h is continuous in the spatial variable, then E possesses the ε-εN-1N property for all constants C>0 (that is, given any ball B, then the constant C of Definition 1 can be taken arbitrarily small, up to choosing ε¯ small enough).

Theorem B (Boundedness).

Assume that there exists a constant M>0 such that

and that E⊆RN is an isoperimetric set for which the ε-εN-1N property holds with an arbitrarily small constant C. Then E is bounded.

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 HN-1-a.e. with its topological boundary. In addition, if h=h⁢(x) is locally α-Hölder for some α∈(0,1], then ∂*⁡E is C1,η, where

Proof of Theorem A.

Let E⊆ℝN be a set of locally finite perimeter, and let B⊆ℝN be a ball such that ℋN-1⁢(B∩∂*⁡E)>0. Since f and h are locally bounded and l.s.c., there exists some constant M>0 such that

Let x∈∂*⁡E∩B be any point of the reduced boundary of E in B. Up to a rotation and a translation, we may assume that x=0 and that νE⁢(x)=(0,1)∈ℝN-1×ℝ.

Step I: Choice of the reference cube and the “good” part G. First of all, we let ρ=ρ⁢(M,N)>0 be a small parameter, only depending on M and on N, which will be chosen later on. As a direct application of the Blow-Up Theorem 2 we obtain the existence of an arbitrarily small constant a>0 such that, calling QN=(-a2,a2)N and Q=(-a2,a2)N-1 the N-dimensional and the (N-1)-dimensional cubes of side a, and letting x=(x′,xN)∈ℝN-1×ℝ one has

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 a=a⁢(M,N,E). Observe that, again in view of the Blow-Up Theorem 2, inside the cube QN one has to expect the set E to be very close to the lower half-cube. As a consequence, a “standard” vertical section of E inside QN should be close to the segment (-a2,0). Writing then for brevity Ex′Q=Ex′∩QN and ∂*⁡Ex′Q=∂*⁡Ex′∩Q, we define then G⊆Q the “good” sections, that is,

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 x′∈Q such that x′∉G. This can happen for different reasons, for instance x′ can belong to the set Γ1 of the sections for which ∂*⁡(Ex′)≠(∂*⁡E)x′, or to the set Γ2 of the sections such that (∂*⁡E)x′ contains at least two points. Let us then call Γ3=Q∖(G∪Γ1∪Γ2), and consider a point x′ in Γ3. We have that ∂*⁡(Ex′)=(∂*⁡E)x′ and that it contains either no point, or a single point. In this second case, this single point either does not belong to (-a⁢ρ,a⁢ρ), or it belongs to (-a⁢ρ,a⁢ρ) and the section Ex′ is the upper part of the segment instead of the lower one, that is, Ex′⊆(-a⁢ρ,a2). In any case, we immediately obtain that for any x′∈Γ3 one has

and by (2.3) we deduce

Instead, Vol’pert Theorem 4 gives that

Since the projection on the first N-1 coordinates is 1-Lipschitz and by definition of Γ2 we have

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 Q=G∪(Γ1∪Γ2∪Γ3), this estimate together with (2.8) and (2.7) implies (2.5). Finally, (2.5) together with (2.9) and (2.10) directly gives (2.6).

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 x′∈ℝN-1 and ℓ>0 let us call Qℓ⁢(x′)⊆ℝN-1 the cube with center in x′, side ℓ, and with sides parallel to the coordinate planes. Let us then fix some c∈Q and some ℓ>0 such that Q2⁢ℓ(c)⊂⊂Q. We aim to show that

To prove this estimate, let us fix a direction 1≤i≤N-1, and for every x′∈Qℓ⁢(c) let us call

We have then

The same estimate is clearly valid replacing Si- with Si+, defined in the same way with yi=xi+ℓ in place of yi=xi-ℓ. Since for every x′∈Qℓ⁢(c) one clearly has ∂⁡Qℓ⁢(x′)⊆⋃i=1N-1Si-⁢(x′)∪Si+⁢(x′), summing the last estimate among all 1≤i≤N-1, we immediately get (2.11).

Step III: Selection of “good” horizontal cubes Qj. We now fix 0≤γ<1N-1 with γ=γ⁢(N,α), whose precise value is given in (2.39). Moreover, we fix a very small constant ε¯>0, which will be precised later, and which depends on a, M and γ, so that in the end ε¯=ε¯⁢(M,N,E,α). Let also 0<ε<ε¯ be given. In this step, we define H=H⁢(γ,N) pairwise disjoint cubes Qj⊆Q, and in the next step we will select one of them to go on with the construction. If γ=0, we only take one cube, that is H⁢(0,N)=1 and the unique cube is Q1=Q itself. If γ>0, instead, we take pairwise disjoint (N-1)-dimensional open cubes Q2⁢ℓ(xj+)⊂⊂Q, where we write for brevity ℓ=a⁢εγ. Notice that it is possible to find 2⁢H of these pairwise disjoint cubes with

For each cube, we can find a point xj∈Qℓ⁢(xj+) such that, calling Qj=Qℓ⁢(xj),

which by (2.11) gives

Keeping in mind that the cubes Q2⁢ℓ⁢(xj+) are pairwise disjoint and contained in Q, as well as (2.10), we deduce that for all 1≤j≤H,

provided that ρ≪1N. In addition, by Vol’pert Theorem 4, we can also assume that for every 1≤j≤H,

Notice that E∩(∂⁡Qj×(-a2,a2)) is contained in the union of 2⁢(N-1) hyperplanes of dimension N-1, so the boundary in the right-hand side above has to be intended as the corresponding union of the (N-2)-dimensional boundaries. Observe that also in the case γ=0 the unique cube Q1=Q satisfies (2.13), which is in fact weaker than (2.4), and without loss of generality we can also assume the validity of (2.14) by Vol’pert Theorem.

Step IV: Choice of one of the horizontal cubes Qj. In this step we select a particular horizontal cube among those defined in the previous step. In particular, we aim to find some 1≤j≤H such that the cubes Qε=Qj and QεN=Qj×(-a2,a2) satisfy

being G the set defined in Step I. Let us start noticing that everything is trivial for the special case γ=0. Indeed, in this case there is nothing to choose since there is only one cube Q1=Q, hence we only have to check the validity of properties (2.15)–(2.18). And in turn, estimates (2.1), (2.2), (2.3) and (2.5) exactly provide the validity of these four inequalities, with even better constants. By the way, in this particular case the cubes Qε=Q and QεN=QN actually do not even depend on ε.

We focus then on the non-trivial case γ>0. First of all, keeping in mind (2.2) and that the cubes Qj are pairwise disjoint we obtain that strictly less than 14 of the H cubes Qj are those for which

hence by (2.12) for more than 34 of the cubes (2.16) is valid. With the very same argument one obtains, for more than 34 of the cubes Qj, also the validity of (2.17) from (2.3) and of (2.18) from (2.5).

The argument to obtain (2.15) is slightly more involved. More precisely, calling A the projection over Q of ∂*⁡E∩QN, for every Borel set V⊆Q we define

It is immediate to notice that ζ is a positive measure and, observing that Q∖A⊆Γ3, from (2.7) and (2.10) we get ζ⁢(Q)≤3⁢ρ⁢aN-1. Hence, arguing as for the other inequalities, we obtain that for more than 34 of the H cubes Qj one has

which implies (2.15). Summarising, we can find at least one index j with 1≤j≤H such that all estimates (2.15)–(2.18) hold true for such an index. For future use we remark that, putting together (2.15) and (2.18), we have

Step V: Definition of σ± and of the modified sets Fδ⊆RN. In this step we define a one-parameter family of sets Fδ; in the next step we will select a suitable δ such that F=Fδ satisfies the volume constraint in (1.1), and then we will check that this set F also satisfies the perimeter constraint in (1.1). To present the construction, we need yet another constant δ¯≪a, which depends on a,M,γ and ε, so ultimately δ¯=δ¯⁢(M,N,E,α,ε). The precise value of δ¯ is given in (2.28). We define the integer

and we take K pairwise disjoint open strips Si=Qε×(σi,σi+δ¯)⊆QεN, with constants a⁢ρ<σi<a2-δ¯ for 1≤i≤K. Notice that this is possible in view of the definition of K, in particular we can do this in such a way that also the closures Si¯ of the strips are pairwise disjoint. Then by (2.16) and (2.17)

Then there exists at least one strip S+:=Si¯, such that

For the sake of notation, we write σ+=σi¯, so that S+=Qε×(σ+,σ++δ¯) and we keep in mind that a⁢ρ<σ+<a2-δ¯.

On the other hand, by (2.16) we can estimate

where the last equality follows from Vol’pert Theorem. In particular, for almost every -a2<t<-a⁢ρ one has that ℋN-1-a.e. point of the hyper-plane ℝN-1×{t} has density either 0 or 1, and

Consequently, we can find another section -a2<σ-<-a⁢ρ for which in addition

For every 0<δ≤δ¯ we define the set Fδ as

It is easy but important to observe how the set Fδ is defined. The difference between E and Fδ is only inside the cylinder Qε×(σ-,σ++δ). In this cylinder, the flat basis Eσ-∩Qε is stretched vertically of a height δ, the high central part E∩(Qε×(σ-,σ+)) is translated vertically of δ, and the short upper part E∩(Qε×(σ+,σ++δ))⊆E∩S+ is simply removed. Keep in mind that E has density either 0 or 1 at ℋN-1-a.e. point of the basis Qε×{σ-}, therefore the vertical stretch of Eσ-∩Qε makes sense.

Step VI: Volume evaluation and choice of the competitor F. In this step we estimate the volume of the sets Fδ defined in the previous step. We will then select a particular one of these sets, which we will simply denote by F, such that

We shall then show that such a set is the one required by the ε-εβ property, by checking the validity of the inequality P⁢(F)≤P⁢(E)+C⁢εβ.

For a given 0<δ<δ¯, let us define

so that

The last term is the easiest to estimate. Indeed, since of course E2-⊆E∩(Qε×(σ+,σ++δ))⊆E∩S+, recalling that 1M<f,h<M in QN, by (2.20) we directly have

Let us now pass to estimate |E+| and |E1-|. To do so, it is convenient to consider separately the vertical sections corresponding to some x′∈G and the other ones. We start by picking some x′∈Qε∩G. Then (E∩Qε)x′ is a vertical segment of the form (-a2,y) with some -a⁢ρ<y<a⁢ρ, so in particular σ-<y<σ+. Therefore, the section (E1-)x′ is empty, while the section (E+)x′ is the segment (y,y+δ). By Fubini and (2.18), we have then

(2.25)

Let us now instead take x′∈Qε∖G, and assume that (x′,xN)∈E+ for some xN∈(σ-,σ++δ). The fact that (x′,xN)∈E+=F∖E implies that (x′,xN)∉E. Instead, the fact that (x′,xN)∈F implies that (x′,y)∈E, where y=max⁡{xN-δ,σ-}. As a consequence, the segment {x}×(xN-δ,xN) intersects ∂*⁡(Ex′). Analogously, if (x′,xN)∈E1-, this means that (x′,xN)∈E while (x′,max⁡{xN-δ,σ-})∉E, so that {x}×(xN-δ,xN) intersects again ∂*⁡(Ex′). Keeping in mind Vol’pert Theorem, we deduce that

where for every set A⊆ℝN we define

Therefore, also by (2.19) we deduce

We are finally in a position to define δ¯ as

Keep in mind that our construction makes sense only if δ¯≪a, which in turn is true as soon as we have chosen ε¯ small enough (recall that ε¯ could be chosen depending on a,M and γ, and that 0≤γ<1N-1). Putting together (2.23), (2.24), (2.25) and (2.27), and recalling (2.28), in the case δ=δ¯ we can now estimate

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 δ=δ¯4⁢M2, (2.23), (2.25) and (2.27), together with (2.28), allow to deduce

again up to choosing ρ small enough. Since of course the measure of Fδ is continuous on δ, by (2.29) and (2.30) we deduce the existence of some δ between δ¯4⁢M2 and δ¯ such that (2.22) holds. We fix then this δ, and from now on we let F=Fδ.

Step VII: Evaluation of P⁢(F). In this step we evaluate P⁢(F). Notice that, in order to obtain the ε-εβ property, we need to show that P⁢(F)≤P⁢(E)+C⁢εβ. Since F equals E outside of the open cylinder 𝒞=Qε×(σ-,σ++δ), the difference between ∂*⁡F and ∂*⁡E is contained in the closed cylinder. We split ∂*⁡F∩𝒞¯ in four parts, namely Ttop+, Tbtm+, Tltr+ and Tint+, where

(2.31)

We also split ∂*⁡E∩𝒞¯ in four parts in a slightly different way as

(2.32)

We now have to carefully consider the above pieces. The easiest thing to notice is that, since by definition of σ- the set E has density either 0 or 1 at ℋN-1-almost every point of Qε×{σ-}, then

so neither E nor F carry any perimeter on the bottom face of the cylinder 𝒞.

Let us now consider Ttop±. We aim to show that, up to ℋN-1-negligible subsets,

where π⁢(x′,xN)=(x′,σ++δ) denotes the projection over the hyperplane {xN=σ++δ}. By the properties of sets of finite perimeter, for ℋN-1-almost every x′∈Qε we have that E has density either 0, or 12, or 1 both at (x′,σ+) and at (x′,σ++δ). If at least one of them is 12, then x′ belongs to the right set in (2.34). If they are both 0 or both 1, then by construction F has also density 0 or 1 at the point (x′,σ++δ), which then does not belong to Ttop+. Let us then assume that one of the two densities is 0 and the other one is 1. Up to discarding an ℋN-1-negligible quantity of points x′, we can assume that Vol’pert Theorem holds for the section Ex′, and the density of Ex′ at the points σ+ and σ++δ is once 0 and once 1. Consequently, ∂*⁡(Ex′) contains some point in the segment (σ+,σ++δ), and then again x′ belongs to the right set in (2.34). Summarising, we proved the inclusion (2.34) up to sets of zero ℋN-1-measure. Keeping in mind the fact that the projection is 1-Lipschitz and (2.20), and noticing that Ttop-⊆∂*⁡E∩S+¯, we have then

To consider the set Tltr+, we can argue similarly, keeping in mind that by (2.14) Vol’pert Theorem is true for ℋN-2-almost each x′∈∂⁡Qε. Indeed, take (x′,xN) where x′ is a point in ∂⁡Qε at which Vol’pert Theorem holds true, and σ-+δ<xN<σ++δ. Up to ℋN-1-negligible subsets, E has density either 0, or 12 or 1 both at (x′,xN) and at (x′,xN-δ). If the densities are both 0 or both 1, then (x′,xN)∉∂*⁡F, while if one density is 0 and the other one is 1 then again there is some point in the segment {x′}×(xN-δ,xN) which belongs to ∂*⁡E. Summarising, up to ℋN-1-negligible subsets

where τ is defined in (2.26). As a consequence, also recalling (2.13) we have

To conclude, we have to consider Tint±. Since the section σ- has been chosen in such a way that Vol’pert Theorem is true, and in the cylinder 𝒞-=Qε×(σ-,σ-+δ) the set F is simply (Eσ-∩Qε)×(σ-,σ-+δ), we readily obtain

Instead, by construction, ∂*⁡F∩(Qε×(σ-+δ,σ++δ)) is nothing else than a vertical translation of height δ of the set ∂*⁡E∩(Qε×(σ-,σ+))=Tint-. In particular, if (x′,xN)∈Tint-, then νF⁢(x′,xN+δ)=νE⁢(x′,xN). Let us call for brevity ξ the vertical translation of a height δ, i.e. ξ⁢(x′,xN)=(x′,xN+δ), so that for y∈Tint- one has νF⁢(ξ⁢(y))=νE⁢(y). By the local α-Hölder assumption on h in the spatial variable, we have a constant C1 such that |h⁢(ξ⁢(y),ν)-h⁢(y,ν)|≤C1⁢δα for every y∈QN, ν∈𝕊N-1. Consequently, by (2.21) and (2.15) we can estimate

Putting together (2.33), (2.35), (2.36) and (2.37), and recalling that δ≤δ¯ together with the definition (2.28) of δ¯, if ε¯ is small enough, we finally derive

for two constants C2,C3 only depending on N,M and a, so actually on N,M and E.

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 0≤γ<1N-1, we here make an optimal choice. Notice that γ↦1-γ is strictly decreasing while γ↦α+γ⁢(N-1)⁢(1-α) is increasing. As 1-γ≥α+γ⁢(N-1)⁢(1-α) for γ=0 while 1-γ<α+γ⁢(N-1)⁢(1-α) for γ=1N-1, the best choice is the unique value γ such that 1-γ=α+γ⁢(N-1)⁢(1-α), i.e.

With this choice, estimate (2.38) becomes P⁢(F)≤P⁢(E)+C⁢εβ where β=β⁢(N,α) is given by (1.2) and C only depends on M,N,E,α and the local Hölder constant of h.

Step IX: The case of a continuous function h. In this step we consider in more details the case α=0, in which “locally α-Hölder” is a synonym of “locally bounded”. In this case, γ=1N and β=N-1N, so in the previous steps for a small 0<ε<ε¯ we already found a set F such that |F|=|E|+ε and P⁢(F)≤P⁢(E)+C⁢εN-1N. In view of applications, for instance Theorem B, it is important to be able to choose the constant C arbitrarily small. More precisely, for any given constant κ>0, one would be interested to have C≤κ up to choosing ε¯ accordingly. In particular, ε¯ should also depend on κ and clearly if κ becomes very small, so must ε¯. It is simple to see that this improvement is false for a generic locally bounded h, see for instance [14]. Here we show that such an improvement is possible if h is continuous in the spatial variable.

Let us assume that α=0 and that h is continuous in the spatial variable. We only need a slight yet fundamental modification of our argument. More precisely, we fix a large constant L=L⁢(M,N,E,κ), to be specified later on, and we possibly decrease, depending on κ and on L, the value of ε¯ found in the previous steps: we let ε′¯=ε¯L. For every 0<ε<ε¯, we call ε′=εL, so that ε′ can be any number in the interval (0,ε′¯). We aim to find some set F, which equals E outside QN, such that

Correspondingly, we also modify the definition of δ¯, which in place of (2.28) is now

As already done immediately after (2.28), we recall that the construction only makes sense with δ¯≪a, and this is again true as soon as ε¯ is small enough. Up to these modifications, we keep everything unchanged in the first five steps. In Step VI, the volume estimates are again true, and in particular (2.29) and (2.30) now read

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 δ¯4⁢M2<δ<δ¯ such that, calling F=Fδ, the equality |F|-|E|=ε′ holds. We have then to bound the perimeter of F, and this will be done by using the estimates of Step VII with only a single modification.

Exactly in Step VII, we divide ∂*⁡F∩𝒞¯ and ∂*⁡E∩𝒞¯ in the parts Ttop±, Tbtm±, Tltr± and Tint± by (2.31) and (2.32). Recall that ℋN-1⁢(Tbtm+∪Tbtm-)=0 by (2.33), while by (2.35) and (2.36), also recalling (2.41) and that γ=1N, we have

where C4 is a constant only depending on M,N and a, so again on M,N and E, and the last step is true as soon as L is large enough and ε¯ is small enough, both depending on M,N,E and κ.

To conclude, we need to evaluate the perimeter contribution of Tint±. We will argue similarly as what already done in estimate (2.37). The only difference is that this time we do not estimate

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 QN. More precisely, we call ω the continuity modulus of h in the spatial variable inside QN, that is

Then, calling ξ, as in Step VII, the vertical translation of height δ, and recalling that δ<δ¯ and (2.41), we have

Putting this estimate inside (2.37), we now get

where as usual C5=C5⁢(M,N,a) and the last estimate is true for ε¯ small enough, as usual depending on M,N,E and κ.

We can finally conclude. Indeed, the first thing to do is to fix L, depending on M,N,E and κ, so large that (2.42) holds true. Once L has been fixed, we fix ε¯ as small as desired, depending on M,N,E,κ and L. Since h is continuous in the spatial variable, the continuity modulus ω⁢(d) goes to 0 as d goes to 0. Hence, if δ¯ is small enough, we have that

Recalling the definition (2.41) of δ¯, we deduce that the above inequality is true as soon as ε¯ is small enough, depending on M,N,L and a. Inserting this last inequality in (2.43) and recalling (2.42), we obtain P⁢(F)≤P⁢(E)+κ⁢ε′⁣N-1N. Summarising, in the case when α=0 but h is continuous in the spatial variable, for every κ>0 we have found some ε′¯>0 depending on M,N,E,κ such that for every 0<ε′<ε′¯ there is a set F with E△F⊂⊂QN satisfying (2.40). The ε-εβ property with constant κ has then been proved, at least for positive values of ε′.

Step X: Conclusion (i.e. the case ε<0). In this last step, we finally conclude the proof. We are left to consider the case ε<0, which is in fact a simple consequence of what we have proved so far. Summarising, we have taken a set E⊆ℝN of locally finite perimeter, a ball B with ℋN-1⁢(∂*⁡E∩B)>0, and a point x∈∂*⁡E∩B, and we found two positive constants C and ε¯ with the property that, for every 0<ε<ε¯, there exists a set F with F△E⊂⊂B and satisfying (1.1), that is

We can apply this result to the set E^=B∖E, which is also a set of locally finite perimeter and whose reduced boundary also have intersection with B of strictly positive ℋN-1-measure, since ∂*⁡E^∩B=∂*⁡E∩B. Then we have two positive constants C^ and ε¯^ such that for every 0<ε<ε¯^, there exists a set F^ such that

Defining then F=(E∖B)∪(B∖F^), we clearly have

In other words, we automatically have the validity of (1.1) also for negative values of ε, up to possibly decreasing the value of ε¯ (resp., increasing the value of C), in case ε¯^ is smaller (resp., C^ is larger). The proof is then concluded. ∎