Rigidity for perimeter inequality under spherical symmetrisation
RRIGIDITY FOR PERIMETER INEQUALITYUNDER SPHERICAL SYMMETRISATION
F. CAGNETTI, M. PERUGINI, AND D. ST ¨OGER
Abstract.
Necessary and sufficient conditions for rigidity of the perimeter inequalityunder spherical symmetrisation are given. That is, a characterisation for the uniqueness(up to orthogonal transformations) of the extremals is provided. This is obtained througha careful analysis of the equality cases, and studying fine properties of the circular sym-metrisation, which was firstly introduced by P´olya in 1950. Introduction
In this paper we study the perimeter inequality under spherical symmetrisation, givingnecessary and sufficient conditions for the uniqueness, up to orthogonal transformations,of the extremals. Perimeter inequalities under symmetrisation have been studied by manyauthors, see for instance [20, 21] and the references therein. In general, we say thatrigidity holds true for one of these inequalities if the set of extremals is trivial. The studyof rigidity can have important applications to show that minimisers of variational problems(or solutions of PDEs) are symmetric.For instance, a crucial step in the proof of the Isoperimetric Inequality given by EnnioDe Giorgi consists in showing rigidity of Steiner’s inequality (see, for instance, [22, Theo-rem 14.4]) for convex sets (see the proof of Theorem I in Section 4 in [16, 17]). After DeGiorgi, an important contribution in the understanding of rigidity for Steiner’s inequalitywas given by Chleb´ık, Cianchi, and Fusco. In the seminal paper [12], the authors givesufficient conditions for rigidity which are much more general than convexity. After that,this result was extended to the case of higher codimensions in [3], where a quantitativeversion of Steiner’s inequality was also given.Then, necessary and sufficient conditions for rigidity (in codimension 1) were given in [9],in the case where the distribution function is a Special Function of Bounded Variation withlocally finite jump set [9, Theorem 1.29]. The anisotropic case has recently been consideredin [26], where rigidity for Steiner’s inequality in the isotropic and anisotropic setting areshown to be equivalent, under suitable conditions. In the Gaussian setting, where the roleof Steiner’s inequality is played by Ehrhard’s inequality (see [15, Section 4.1]), necessaryand sufficient conditions for rigidity are given in [10], by making use of the notion ofessential connectedness [10, Theorem 1.3]. Finally, in the smooth case, sufficient conditionsfor rigidity are given in [24, Proposition 5], for a general class of symmetrisations in warpedproducts.The main motivation for the study of the spherical symmetrisation is that it can be usedto understand the symmetry properties of the solutions of PDEs and variational problems,when the radial symmetry has been ruled out. Moreover, some well established methods(as for instance the moving plane method, see [29, 19]) rely on convexity properties of thedomain which fail, for instance, when one deals with annuli.In particular, in many applications minimisers of variational problems and solutions ofPDEs turn out to be foliated Schwarz symmetric . Roughly speaking, a function u : R n → R is foliated Schwarz symmetric if one can find a direction p ∈ S n − such that u only dependson | x | and on the polar angle α = arccos(ˆ x · p ), and u is non increasing with respect to a r X i v : . [ m a t h . A P ] A p r (here ˆ x := x/ | x | , and | · | denotes the Euclidean norm in R n ). We direct the interestedreader to [4, 5, 6, 31] and the references therein for more information.1.1. Spherical Symmetrisation.
To the best of our knowledge, the spherical symmetri-sation was first introduced by P´olya in [27], in the case n = 2 and in the smooth setting.Let n ∈ N with n ≥
2. For each r > x ∈ R n , we denote by B ( x, r ) the open ballof R n of radius r centred at x , by ω n the ( n -dimensional) volume of the unit ball, and wewrite B ( r ) for B (0 , r ). Moreover, e , . . . , e n stand for the vectors of the canonical basis of R n . Given a set E ⊂ R n and r >
0, we define the spherical slice E r of E with respect to ∂B ( r ) as E r := E ∩ ∂B ( r ) = { x ∈ E : | x | = r } . Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function. We say that E is spherically v -distributed if v ( r ) = H n − ( E r ) , for H -a.e. r ∈ (0 , ∞ ) , (1.1)where H k denotes the k -dimensional Hausdorff measure of R n , 1 ≤ k ≤ n . Note that, inorder v to be an admissible distribution, one needs v ( r ) ≤ H n − ( ∂B ( r )) = nω n r n − for H -a.e. r ∈ (0 , ∞ ) . (1.2)In the following, as usual, we set S n − = ∂B (1). For every x, y ∈ S n − , the geodesicdistance between x and y is given bydist S n − ( x, y ) := arccos( x · y ) . Let r > p ∈ S n − , and β ∈ [0 , π ] be fixed. The open geodesic ball (or spherical cap ) ofcentre rp and radius β is the set B β ( rp ) := { x ∈ ∂B ( r ) : dist S n − (ˆ x, p ) < β } . The ( n − B β ( rp ) can be explicitly calculated, and isgiven by H n − ( B β ( rp )) = ( n − ω n − r n − ˆ β (sin τ ) n − dτ. The expression above shows that the function β (cid:55)→ H n − ( B β ( rp )) is strictly increasingfrom [0 , π ] to [0 , nω n r n − ]. Therefore, if v : (0 , ∞ ) → [0 , ∞ ) is a measurable functionsatisfying (1.2), and E ⊂ R n is a spherically v -distributed set, there exists only one(defined up to a subset of zero H -measure) measurable function α v : (0 , ∞ ) → [0 , π ]satisfying v ( r ) = H n − ( B α v ( r ) ( re )) for H -a.e. r ∈ (0 , ∞ ) . (1.3)Among all the spherically v -distributed sets of R n , we denote by F v the one whose sphericalslices are open geodesic balls centred at the positive e axis., i.e. F v := { x ∈ R n \ { } : dist S n − (ˆ x, e ) < α v ( | x | ) } , see Figure 1.1. Before stating our results, it will be convenient to recall some basic notionsabout sets of finite perimeter.1.2. Basic notions on sets of finite perimeter.
Let E ⊂ R n be a measurable set, andlet t ∈ [0 , E ( t ) the set of points of density t of E , given by E ( t ) := (cid:26) x ∈ R n : lim ρ → + H n ( E ∩ B ( x, ρ )) ω n ρ n = t (cid:27) . The essential boundary of E is then defined as ∂ e E := E \ ( E (1) ∪ E (0) ) . F v α v ( r ) Er x x x E x x x x x x x x x Figure 1.1.
A pictorial idea of the spherical symmetral F v of a v -distributed set E , in the case n = 3.Moreover, if A ⊂ R n is any Borel set, we define the perimeter of E relative to A as theextended real number given by P ( E ; A ) := H n − ( ∂ e E ∩ A ) , and we set P ( E ) := P ( E ; R n ). When E is a set with smooth boundary, it turns out that ∂ e E = ∂E , and the perimeter of E agrees with the usual notion of ( n − ∂E .If P ( E ) < ∞ , it is possible to define the reduced boundary ∂ ∗ E of E . This has theproperty that ∂ ∗ E ⊂ ∂ e E , H n − ( ∂ e E \ ∂ ∗ E ) = 0, and is such that for every x ∈ ∂ ∗ E there exists the measure theoretic outer unit normal ν E ( x ) of ∂ ∗ E at x , see Section 2. If x ∈ ∂ ∗ E , it will be convenient to decompose ν E ( x ) as ν E ( x ) = ν E ⊥ ( x ) + ν E (cid:107) ( x ) , where ν E ⊥ ( x ) := ( ν E ( x ) · ˆ x )ˆ x and ν E (cid:107) ( x ) are the radial and tangential component of ν E ( x )along ∂B ( | x | ), respectively. In the following, we will use the diffeomorphism Φ : (0 , ∞ ) × S n − → R n \ { } defined asΦ( r, ω ) := rω for every ( r, ω ) ∈ (0 , ∞ ) × S n − . .3. Perimeter Inequality under spherical symmetrisation.
Our first result showsthat the spherical symmetrisation does not increase the perimeter, and gives some neces-sary conditions for equality cases. In our analysis we require the set F v (or, equivalently,any spherically v -distributed set) to have finite volume. This is not restrictive. Indeed, if F v has finite perimeter but infinite volume, we can consider the complement R n \ F v which,by the relative isoperimetric inequality, has finite volume. This change corresponds to con-sidering the complementary distribution function r (cid:55)→ nω n r n − − v ( r ), and the sphericalsymmetrisation with respect to the axis − e . Theorem 1.1.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) , andlet E ⊂ R n be a spherically v -distributed set of finite perimeter and finite volume. Then, v ∈ BV (0 , ∞ ) . Moreover, F v is a set of finite perimeter and P ( F v ; Φ( B × S n − )) ≤ P ( E ; Φ( B × S n − )) , (1.4) for every Borel set B ⊂ (0 , ∞ ) .Finally, if P ( E ) = P ( F v ) , then for H -a.e. r ∈ { < α v < π } : (a) E r is H n − -equivalent to a spherical cap and H n − ( ∂ ∗ ( E r )∆( ∂ ∗ E ) r ) = 0 ; (b) the functions x (cid:55)→ ν E ( x ) · ˆ x and x (cid:55)→ | ν E (cid:107) | ( x ) are constant H n − -a.e. in ( ∂ ∗ E ) r . The result above shows that the perimeter inequality holds on a local level, providedone considers sets of the type Φ( B × S n − ), with B ⊂ (0 , ∞ ) Borel. Inequality (1.4) isvery well known in the literature. In the special case n = 2, a short proof was given byP´olya in [27]. In the general n -dimensional case with B = (0 , ∞ ) the result is stated in[25, Theorem 6.2]), but the proof is only sketched (see also [23] and [24, Proposition 3 andRemark 4]). As mentioned by Morgan and Pratelli in [25], certain parts of the proof of(1.4) follow the general lines of analogous results in the context of Steiner symmetrisation(see, for instance, [12, Lemma 3.4] and [3, Theorem 1.1]). There are, however, non trivialtechnical difficulties that arise when one deals with the spherical symmetrisation. For thisreason, we give a detailed proof of Theorem 1.1.We start by introducing radial and tangential components of a Radon measure, seeSection 3.1. These turn out to be useful tools which allow to prove several preliminaryresults. Moreover, since we are dealing with a symmetrisation of codimension n −
1, weneed to pay attention to some delicate effects that are not usually observed when thecodimension is 1 (as, for instance, in [12]). Indeed, a crucial role is played by the measure λ E given by: λ E ( B ) := ˆ ∂ ∗ E ∩ Φ( B × S n − ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H n − ( x ) , (1.5)for every Borel set B ⊂ (0 , ∞ ). When n = 2, it turns out that λ E is singular with respectto the Lebesgue measure in (0 , ∞ ). However, for n > λ E containsa non trivial absolutely continuous part, see Remark 3.9. This requires some extra carewhile proving inequality (1.4). A similar phenomenon has already been observed in [3], inthe study of the Steiner symmetrisation of codimension higher than 1. Higher codimensioneffects play an important role also in the study of rigidity, as explained below.1.4. Rigidity of the Perimeter Inequality.
Given v : (0 , ∞ ) → [0 , ∞ ) measurable,satisfying (1.2), and such that F v is a set of finite perimeter and finite volume, we define N ( v ) as the class of extremals of (1.4): N ( v ) := { E ⊂ R n : E is spherically v -distributed and P ( E ) = P ( F v ) } . Note that, by definition of F v , and by the invariance of the perimeter under rigid trans-formations, every time we apply an orthogonal transformation to F v we obtain a set that elongs to N ( v ), i.e.: N ( v ) ⊃ { E ⊂ R n : H n ( E ∆( R F v )) = 0 for some R ∈ O ( n ) } , where ∆ denotes the symmetric difference of sets and O ( n ) is the set of orthogonal trans-formations in R n . We would like to understand when also the opposite inclusion is satisfied,that is, when the class of extremals of (1.4) is just given by rotated copies of F v . We willsay that rigidity holds true for inequality (1.4) if N ( v ) = { E ⊂ R n : H n ( E ∆( R F v )) = 0 for some R ∈ O ( n ) } . ( R )In order to explain which conditions we should expect in order ( R ) to be true, let us firstgive some examples.Figure 1.2 shows a set E ∈ N ( v ) that cannot be obtained by applying a single orthogonaltransformation to F v . This is due to the fact that the set { < α v < π } is disconnected˜ r x x E x x ˜ rF v Figure 1.2.
Rigidity ( R ) fails, since the set { < α v < π } is disconnectedby a point ˜ r ∈ (0 , ∞ ) such that α v (˜ r ) = 0.by a point ˜ r satisfying α v (˜ r ) = 0. A similar situation happens when { < α v < π } isdisconnected by points belonging to the set { α v = π } , see Figure 1.3.ˆ r x x E ˆ r x x F v Figure 1.3.
The set E above cannot be obtained by applying an orthog-onal transformation around the origin to the set F v shown in the right,therefore rigidity ( R ) fails. This happens because the set { < α v < π } isdisconnected by a point ˆ r ∈ (0 , ∞ ) such that α v (ˆ r ) = π .One possibility to avoid such a situation could be to request the set { < α v < π } tobe an interval. However, this condition depends on the representative chosen for α v , whilethe perimeters of the sets E and F v don’t. Indeed, in Figure 1.2 one could modify α v just at the point ˜ r , in such a way that { < α v < π } becomes an interval. Nevertheless,rigidity still fails, see Figure 1.4. o formulate a condition which is independent on the chosen representative, we considerthe approximate liminf and the approximate limsup of α v , which we denote by α ∧ v and α ∨ v ,respectively (see Section 2). These two functions are defined at every point r ∈ (0 , ∞ ) andsatisfy α ∧ v ≤ α ∨ v . In addition, they do not depend on the representative chosen for α v , and α ∧ v = α ∨ v = α v H -a.e. in (0 , ∞ ). The condition that we will impose is then the following: { < α ∧ v ≤ α ∨ v < π } is a (possibly unbounded) interval. (1.6)One can check that, in the example given in Figure 1.4 this condition fails, since α ∧ v (˜ r ) = α ∨ v (˜ r ) = 0. ˜ r x x E x x ˜ rF v Figure 1.4.
Modifying the function α v given in Figure 1.2 at the point˜ r , we can make sure that { < α v < π } is an open connected interval.However, rigitidy still fails.Let us show that, even imposing (1.6), rigidity can still be violated. In the examplegiven in Figure 1.5, there is some radius r ∈ { < α ∧ v ≤ α ∨ v < π } such that the boundaryof F v contains a non trivial subset of ∂B ( r ). In this way, it is possible to rotate a propersubset of F v around the origin, without affecting the perimeter. Note that at each pointof the set ∂ ∗ F v ∩ ∂B ( r ) the exterior normal ν F v is parallel to the radial direction. To ruleout the situation described in Figure 1.5, we will impose the following condition: H n − ( { x ∈ ∂ ∗ F v : ν F v (cid:107) ( x ) = 0 and | x | ∈ { < α ∧ v ≤ α ∨ v < π } ) = 0 . (1.7)Note that, from Theorem 1.1 and identity (1.3), it follows that in general we only have α v ∈ BV loc (0 , ∞ ). However, it turns out that (1.7) is equivalent to ask that α v is W , inthe interior of { < α ∧ v ≤ α ∨ v < π } , see Proposition 5.3. E x rx x x F v r Figure 1.5.
An example in which rigidity fails. In this case, the tangentialpart of ∂ ∗ F v gives a non trivial contribution to P ( F v ). This allows to slidea proper subset of F v around the origin, without modifying the perimeter. ur main result shows that the two conditions above give a complete characterisationof rigidity for inequality (1.4) (below, ˚ I stands for the interior of the set I ). Theorem 1.2.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) such that F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3) . Then, thefollowing two statements are equivalent: (i) ( R ) holds true; (ii) { < α ∧ v ≤ α ∨ v < π } is a (possibly unbounded) interval I , and α v ∈ W , (˚ I ) . Let us point out that, although similar results in the context of Steiner and Ehrhard’sinequalities already appeared in [9, 10], the proof of Theorem 1.2 cannot simply use previ-ous ideas, especially in the implication (i) = ⇒ (ii). We cannot rely, as in [9], on a generalformula for the perimeter of sets E satisfying equality in (1.4). Instead, we exhibit explicitcounterexamples to rigidity, whenever one of the assumptions in (ii) fails. This requires acareful analysis of the transformations that one can apply to the set F v , without modifyingits perimeter. This turns out to be non trivial, especially if one assumes α v to have a nonzero Cantor part (see Proposition 8.4).Also the proof of the implication (ii) = ⇒ (i) presents some difficulties. In the context ofSteiner symmetrisation, this has been proved in [12, Theorem 1.3] and [3, Theorem 1.2],for codimension 1 and for every codimension, respectively. In the smooth case, a proof isgiven in [24, Proposition 5], for the general class of symmetrisations in warped products.For the spherical setting without any smoothness assumption, this implication has alreadybeen stated in [25, Theorem 6.2], but the proof is only sketched. A rigorous proof of thisfact turns out to be more delicate than one would expect, and relies on the following result. Lemma 1.3.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) such that F v is a set of finite perimeter and finite volume. Let E ⊂ R n be a spherically v -distributedset, and let I ⊂ (0 , + ∞ ) be a Borel set. Assume that H n − (cid:16)(cid:110) x ∈ ∂ ∗ E ∩ Φ( I × S n − ) : ν E (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . (1.8) Then, H n − (cid:16)(cid:110) x ∈ ∂ ∗ F v ∩ Φ( I × S n − ) : ν F v (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . (1.9) Viceversa, let (1.9) be satisfied, and suppose that P ( E ; Φ( I × S n − )) = P ( F v ; Φ( I × S n − )) .Then, (1.8) holds true. A direct proof of Lemma 1.3 does not seem to be obvious, due to the fact that, aspointed out above, the measure λ E defined in (1.5) can have an absolutely continuouspart when n >
2. In the context of Steiner symmetrisation of higher codimension, a resultplaying the role of Lemma 1.3 (see [3, Proposition 3.6]) is proved using the fact that thestatement holds true in codimension 1, see [12, Proposition 4.2]. For this reason, we areled to consider the circular symmetrisation , which is the codimension 1 version of thespherical symmetrisation, and was originally introduced by P´olya in the case n = 3 (see[27]). Note that, when n = 2, spherical and circular symmetrisation coincide.1.5. Circular Symmetrisation.
In order to introduce the circular symmetrisation, letus first observe how the spherical symmetrisation operates on a given set E , in the specialcase n = 2. In this situation, for each r > E with the circle ∂B ( r ) ofradius r centred at the origin. Then, the symmetric set F v is obtained by centring, foreach r >
0, an open circumference arc of length H ( E ∩ ∂B ( r )) at the point re . When n > E with parallel planes, andthen by symmetrising it (in each plane) with the procedure just described. Note that, inthis case, one needs to specify both the direction along which the open arcs are centred,and the direction along which the slicing through planes is performed. et us then choose an ordered pair of orthogonal directions in R n , which we will assumeto be ( e , e ) (we will be centring open circumference arcs along e , while we will beslicing the set E with parallel planes that are orthogonal to e ). In the following, foreach x = ( x , . . . , x n ) ∈ R n , we will write x = ( x , x (cid:48) ), where x = ( x , x ) ∈ R and x (cid:48) = ( x , . . . , x n ) ∈ R n − . When x (cid:54) = 0, we set ˆ x := x / | x | . For each given z (cid:48) ∈ R n − , we denote by Π z (cid:48) the two-dimensional plane defined byΠ z (cid:48) := { x = ( x , x (cid:48) ) ∈ R × R n − : x (cid:48) = z (cid:48) } . Given a set E ⊂ R n and ( r, z (cid:48) ) ∈ (0 , ∞ ) × R n − , we define the circular slice E ( r,z (cid:48) ) of E with respect to ∂B ((0 , z (cid:48) ) , r ) ∩ Π z (cid:48) as E ( r,z (cid:48) ) := E ∩ ∂B ((0 , z (cid:48) ) , r ) ∩ Π z (cid:48) = { x = ( x , x (cid:48) ) ∈ E : x (cid:48) = z (cid:48) and | x | = r } . Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function. We say that E is circularly (cid:96) -distributed if (cid:96) ( r, x (cid:48) ) = H ( E ( r,x (cid:48) ) ) , for H n − -a.e. ( r, x (cid:48) ) ∈ (0 , ∞ ) × R n − . If (cid:96) is a circular distribution, then we have (cid:96) ( r, x (cid:48) ) ≤ H ( ∂B ((0 , x (cid:48) ) , r ) ∩ Π x (cid:48) ) = 2 πr for H n − -a.e. ( r, x (cid:48) ) ∈ (0 , ∞ ) × R n − . (1.10)Among all the sets in R n that are circularly (cid:96) -distributed, we denote by F (cid:96) the one whosecircular slices are open circumference arcs centred at the positive e axis. That is, we set F (cid:96) := (cid:26) ( x , x (cid:48) ) ∈ R n \ { x = 0 } : dist S (ˆ x , e ) < r (cid:96) ( r, x (cid:48) ) (cid:27) . In the following, we introduce the diffeomorphism Φ : (0 , ∞ ) × R n − × S → R n \{ ˆ x = 0 } given by Φ ( r, x (cid:48) , ω ) := ( rω, x (cid:48) ) for every ( r, x (cid:48) , ω ) ∈ (0 , ∞ ) × R n − × S . Moreover, for every x ∈ ∂ ∗ E we write ν E ( x ) = ( ν E ( x ) , ν Ex (cid:48) ( x )), where ν E ( x ) = ( ν E ( x ) , ν E ( x ))and ν Ex (cid:48) ( x ) = ( ν E ( x ) , . . . , ν En ( x )). Then, we further decompose ν E ( x ) as ν E ( x ) = ν E ⊥ ( x ) + ν E (cid:107) ( x ) , where ν E ⊥ ( x ) := ( ν E ( x ) · ˆ x )ˆ x and ν E (cid:107) ( x ) := ν E ( x ) − ν E ⊥ ( x ). We can now state aresult that plays the role of Theorem 1.1 for the circular symmetrisation. Theorem 1.4.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) ,and let E ⊂ R n be a circularly (cid:96) -distributed set of finite perimeter and finite volume. Then, (cid:96) ∈ BV loc ((0 , ∞ ) × R n − ) . Moreover, F (cid:96) is a set of finite perimeter and P ( F (cid:96) ; Φ ( B × S )) ≤ P ( E ; Φ ( B × S )) , (1.11) for every Borel set B ⊂ (0 , ∞ ) × R n − .Finally, if P ( E ) = P ( F (cid:96) ) , then for H n − -a.e. ( r, x (cid:48) ) ∈ (0 , ∞ ) × R n − : (a) E ( r,x (cid:48) ) is H -equivalent to a circular arc and ∂ ∗ ( E ( r,x (cid:48) ) ) = ( ∂ ∗ E ) ( r,x (cid:48) ) ; (b) the three functions x (cid:55)−→ ν E ( x ) · ˆ x , x (cid:55)−→ | ν E (cid:107) | ( x ) , x (cid:55)−→ ν Ex (cid:48) ( x ) , are constant in ( ∂ ∗ E ) ( r,x (cid:48) ) . In the smooth setting and in the case n = 3, inequality (1.11) was proved by P´olya. Thefollowing result is the counterpart of Lemma 1.3 in the context of circular symmetrisation. emma 1.5. Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) such that F (cid:96) is a set of finite perimeter and finite volume. Let E ⊂ R n be a circularly (cid:96) -distributed set, and let I ⊂ (0 , ∞ ) × R n − be a Borel set. Assume that H n − (cid:16)(cid:110) x ∈ ∂ ∗ E ∩ Φ( I × S ) : ν E (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . (1.12) Then, H n − (cid:16)(cid:110) x ∈ ∂ ∗ F (cid:96) ∩ Φ( I × S ) : ν F (cid:96) (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . (1.13) Viceversa, let (1.13) be satisfied, and suppose that P ( E ; Φ( I × S )) = P ( F (cid:96) ; Φ( I × S )) .Then, (1.12) holds true. Once Lemma 1.5 is established, we can show Lemma 1.3 through a slicing argument.Finally, the proof of (ii) = ⇒ (i) is concluded by showing that, if E satisfies equality in(1.4), the function associating to every r ∈ (0 , ∞ ) the center of E r (see (7.1)) is W , and,ultimately, constant (see Section 7).The paper is divided as follows. Section 2 contains basic results of Geometric MeasureTheory that are extensively used in the following. In Section 3 we give the setting ofthe problem and introduce useful tools to deal with the spherical framework. Section 4is devoted to the study of the properties of the functions v and ξ v , while Theorem 1.1 isproven in Section 5. Important properties of the circular symmetrisation are discussed inSection 6, where we also give the proof of Lemma 1.3. The implications (ii) = ⇒ (i) and(i) = ⇒ (ii) of Theorem 1.2 are proven in Section 7 and Section 8, respectively.2. Basic notions of Geometric Measure Theory
In this section we introduce some tools from Geometric Measure Theory. The interestedreader can find more details in the monographs [2, 18, 22, 30]. For n ∈ N , we denote with S n − the unit sphere of R n , i.e. S n − = { x ∈ R n : | x | = 1 } , where | · | stands for the Euclidean norm, and we set R n := R n \ { } . For every x ∈ R n , wewrite ˆ x := x/ | x | for the radial versor of x . We denote by e , . . . , e n the canonical basis in R n , and for every x, y ∈ R n , x · y stands for the standard scalar product in R n between x and y . For every r > x ∈ R n , we denote by B ( x, r ) the open ball of R n with radius r centred at x . In the special case x = 0, we set B ( r ) := B (0 , r ). In the following, we willoften make use of the diffeomorphism Φ : (0 , ∞ ) × S n − → R n defined asΦ( r, ω ) := rω for every ( r, ω ) ∈ (0 , ∞ ) × S n − . For x ∈ R n and ν ∈ S n − , we will denote by H + x,ν and H − x,ν the closed half-spaces whoseboundaries are orthogonal to ν : H + x,ν := (cid:110) y ∈ R n : ( y − x ) · ν ≥ (cid:111) , (2.1) H − x,ν := (cid:110) y ∈ R n : ( y − x ) · ν ≤ (cid:111) . If 1 ≤ k ≤ n , we denote by H k the k -dimensional Hausdorff measure in R n . If { E h } h ∈ N is a sequence of Lebesgue measurable sets in R n with finite volume, and E ⊂ R n is alsomeasurable with finite volume, we say that { E h } h ∈ N converges to E as h → ∞ , and write E h → E , if H n ( E h ∆ E ) → h → ∞ . In the following, we will denote by χ E thecharacteristic function of a measurable set E ⊂ R n . .1. Density points.
Let E ⊂ R n be a Lebesgue measurable set and let x ∈ R n . Theupper and lower n -dimensional densities of E at x are defined as θ ∗ ( E, x ) := lim sup r → + H n ( E ∩ B ( x, r )) ω n r n , θ ∗ ( E, x ) := lim inf r → + H n ( E ∩ B ( x, r )) ω n r n , respectively. It turns out that x (cid:55)→ θ ∗ ( E, x ) and x (cid:55)→ θ ∗ ( E, x ) are Borel functions thatagree H n -a.e. on R n . Therefore, the n -dimensional density of E at xθ ( E, x ) := lim r → + H n ( E ∩ B ( x, r )) ω n r n , is defined for H n -a.e. x ∈ R n , and x (cid:55)→ θ ( E, x ) is a Borel function on R n . Given t ∈ [0 , E ( t ) := { x ∈ R n : θ ( E, x ) = t } . By the Lebesgue differentiation theorem, the pair { E (0) , E (1) } is a partition of R n , up toa H n -negligible set. The set ∂ e E := R n \ ( E (0) ∪ E (1) ) is called the essential boundary of E .2.2. Rectifiable sets.
Let 1 ≤ k ≤ n , k ∈ N . If A, B ⊂ R n are Borel sets we say that A ⊂ H k B if H k ( B \ A ) = 0, and A = H k B if H k ( A ∆ B ) = 0, where ∆ denotes the symmetricdifference of sets. Let M ⊂ R n be a Borel set. We say that M is countably H k -rectifiable if there exist Lipschitz functions f h : R k → R n ( h ∈ N ) such that M ⊂ H k (cid:83) h ∈ N f h ( R k ).Moreover, we say that M is locally H k -rectifiable if H k ( M ∩ K ) < ∞ for every compactset K ⊂ R n , or, equivalently, if H k (cid:120) M is a Radon measure on R n .A Lebesgue measurable set E ⊂ R n is said of locally finite perimeter in R n if there existsa R n -valued Radon measure µ E , called the Gauss–Green measure of E , such that ˆ E ∇ ϕ ( x ) dx = ˆ R n ϕ ( x ) dµ E ( x ) , ∀ ϕ ∈ C c ( R n ) , where C c ( R n ) denotes the class of C functions in R n with compact support. The relativeperimeter of E in A ⊂ R n is then defined by setting P ( E ; A ) := | µ E | ( A ) for any Borelset A ⊂ R n . The perimeter of E is then defined as P ( E ) := P ( E ; R n ). If P ( E ) < ∞ , wesay that E is a set of finite perimeter in R n . The reduced boundary of E is the set ∂ ∗ E ofthose x ∈ R n such that ν E ( x ) = lim r → + µ E ( B ( x, r )) | µ E | ( B ( x, r )) exists and belongs to S n − . The Borel function ν E : ∂ ∗ E → S n − is called the measure-theoretic outer unit normal to E . If E is a set of locally finite perimeter, it is possible to show that ∂ ∗ E is a locally H n − -rectifiable set in R n [22, Corollary 16.1], with µ E = ν E H n − ∂ ∗ E , and ˆ E ∇ ϕ ( x ) dx = ˆ ∂ ∗ E ϕ ( x ) ν E ( x ) d H n − ( x ) , ∀ ϕ ∈ C c ( R n ) . Thus, P ( E ; A ) = H n − ( A ∩ ∂ ∗ E ) for every Borel set A ⊂ R n . If E is a set of locally finiteperimeter, it turns out that ∂ ∗ E ⊂⊂ E (1 / ⊂ ∂ e E .
Moreover,
Federer’s theorem holds true (see [2, Theorem 3.61] and [22, Theorem 16.2]): H n − ( ∂ e E \ ∂ ∗ E ) = 0 , thus implying that the essential boundary ∂ e E of E is locally H n − -rectifiable in R n . .3. General facts about measurable functions.
Let f : R n → R be a Lebesguemeasurable function. We define the approximate upper limit f ∨ ( x ) and the approximatelower limit f ∧ ( x ) of f at x ∈ R n as f ∨ ( x ) = inf (cid:110) t ∈ R : x ∈ { f > t } (0) (cid:111) , (2.2) f ∧ ( x ) = sup (cid:110) t ∈ R : x ∈ { f < t } (0) (cid:111) . (2.3)We observe that f ∨ and f ∧ are Borel functions that are defined at every point of R n , withvalues in R ∪ {±∞} . Moreover, if f : R n → R and f : R n → R are measurable functionssatisfying f = f H n -a.e. on R n , then f ∨ = f ∨ and f ∧ = f ∧ everywhere on R n . Wedefine the approximate discontinuity set S f of f as S f := { f ∧ < f ∨ } . Note that, by the above considerations, it follows that H n ( S f ) = 0. Although f ∧ and f ∨ may take infinite values on S f , the difference f ∨ ( x ) − f ∧ ( x ) is well defined in R ∪ {±∞} for every x ∈ S f . Then, we can define the approximate jump [ f ] of f as the Borel function[ f ] : R n → [0 , ∞ ] given by[ f ]( x ) := (cid:40) f ∨ ( x ) − f ∧ ( x ) , if x ∈ S f , , if x ∈ R n \ S f . Let A ⊂ R n be a Lebesgue measurable set. We say that t ∈ R ∪ {±∞} is the approximatelimit of f at x with respect to A , and write t = ap lim( f, A, x ), if θ (cid:16) {| f − t | > ε } ∩ A ; x (cid:17) = 0 , ∀ ε > , ( t ∈ R ) ,θ (cid:16) { f < M } ∩ A ; x (cid:17) = 0 , ∀ M > , ( t = + ∞ ) ,θ (cid:16) { f > − M } ∩ A ; x (cid:17) = 0 , ∀ M > , ( t = −∞ ) . We say that x ∈ S f is a jump point of f if there exists ν ∈ S n − such that f ∨ ( x ) = ap lim( f, H + x,ν , x ) , f ∧ ( x ) = ap lim( f, H − x,ν , x ) . If this is the case, we say that ν f ( x ) := ν is the approximate jump direction of f at x .If we denote by J f the set of approximate jump points of f , we have that J f ⊂ S f and ν f : J f → S n − is a Borel function.2.4. Functions of bounded variation.
Let f : R n → R be a Lebesgue measurablefunction, and let Ω ⊂ R n be open. We define the total variation of f in Ω as | Df | (Ω) = sup (cid:110) ˆ Ω f ( x ) div T ( x ) dx : T ∈ C c (Ω; R n ) , | T | ≤ (cid:111) , where C c (Ω; R n ) is the set of C functions from Ω to R n with compact support. We alsodenote by C c (Ω; R n ) the class of all continuous functions from Ω to R n . Analogously, forany k ∈ N , the class of k times continuously differentiable functions from Ω to R n is denotedby C kc (Ω; R n ). We say that f belongs to the space of functions of bounded variations, f ∈ BV (Ω), if | Df | (Ω) < ∞ and f ∈ L (Ω). Moreover, we say that f ∈ BV loc (Ω) if f ∈ BV (Ω (cid:48) ) for every open set Ω (cid:48) compactly contained in Ω. Therefore, if f ∈ BV loc ( R n )the distributional derivative Df of f is an R n -valued Radon measure. In particular, E is aset of locally finite perimeter if and only if χ E ∈ BV loc ( R n ). If f ∈ BV loc ( R n ), one can writethe Radon–Nykodim decomposition of Df with respect to H n as Df = D a f + D s f , where D s f and H n are mutually singular, and where D a f (cid:28) H n . We denote the density of D a f with respect to H n by ∇ f , so that ∇ f ∈ L (Ω; R n ) with D a f = ∇ f d H n . Moreover, for H n -a.e. x ∈ R n , ∇ f ( x ) is the approximate differential of f at x . If f ∈ BV loc ( R n ), then S f s countably H n − -rectifiable. Moreover, we have H n − ( S f \ J f ) = 0, [ f ] ∈ L loc ( H n − (cid:120) J f ),and the R n -valued Radon measure D j f defined as D j f = [ f ] ν f d H n − (cid:120) J f , is called the jump part of Df . If we set D c f = D s f − D j f , we have that Df = D a f + D j f + D c f . The R n -valued Radon measure D c f is called the Cantorian part of Df , andit is such that | D c f | ( M ) = 0 for every M ⊂ R n which is σ -finite with respect to H n − .In the special case n = 1, if ( a, b ) ⊂ R is an open (possibly unbounded) interval, every f ∈ BV (( a, b )) can be written as f = f a + f j + f c , (2.4)where f ∈ W , (( a, b )), f j is a jump function (i.e. Df = D j f ) and f c is a Cantor function(i.e. Df = D c f ), see [2, Corollary 3.33]. Moreover, if f j = 0 (or, more in general, if f is a good representative , see [2, Theorem 3.28]), the total variation of Df can be obtained as | Df | ( a, b ) = sup (cid:40) N (cid:88) i =1 | f ( x i +1 ) − f ( x i ) | : a < x < x < . . . < x N < b (cid:41) , (2.5)where the supremum runs over all N ∈ N , and over all the possible partitions of ( a, b ) with a < x < x < . . . < x N < b . When n = 1, we will often write f (cid:48) instead of ∇ f .3. Setting of the problem and preliminary results
In this section we give the notation for the chapter, and we introduce some results thatwill be extensively used later. For every x, y ∈ S n − , the geodesic distance between x and y is given by dist S n − ( x, y ) := arccos( x · y ) . We recall that the geodesic distance satisfies the triangle inequality:dist S n − ( x, y ) ≤ dist S n − ( x, z ) + dist S n − ( z, y ) for every x, y, z ∈ S n − . Let r > p ∈ S n − and β ∈ [0 , π ] be fixed. The open geodesic ball (or spherical cap ) ofcentre rp and radius β is the set B β ( rp ) := { x ∈ ∂B ( r ) : dist S n − (ˆ x, p ) < β } . Note in the extreme cases β = 0 and β = π we have B ( rp ) = ∅ and B π ( rp ) = ∂B ( r ) \{− rp } , respectively. Accordingly, the geodesic sphere of centre rp and radius β is theboundary of B β ( rp ), which is given by S β ( rp ) := { x ∈ ∂B ( r ) : dist S n − (ˆ x, p ) = β } . The ( n − n − H n − ( B β ( rp )) = ( n − ω n − r n − ˆ β (sin τ ) n − dτ, (3.1) H n − ( S β ( rp )) = ( n − ω n − r n − (sin β ) n − . (3.2)Let E ⊂ R n be a measurable set. For every r >
0, we define the spherical slice of radius r of E as the set E r := E ∩ ∂B ( r ) = { x ∈ ∂B ( r ) : x ∈ E } . Let v : (0 , ∞ ) → [0 , ∞ ) be a Lebesgue measurable function, and let E ⊂ R n be a measur-able set in R n . We say that E is spherically v -distributed if v ( r ) = H n − ( E r ) , for H -a.e. r ∈ (0 , ∞ ) . f E is spherically v -distributed, we can define the function ξ v ( r ) := v ( r ) r n − = H n − ( E r ) r n − , for every r ∈ (0 , ∞ ) . (3.3)Note that H n − ( B π ) = H n − ( S n − ) = nω n , so that0 ≤ ξ v ( r ) ≤ nω n , for every r ∈ (0 , ∞ ) . (3.4)From (3.1), it follows that the function F : [0 , π ] → [0 , nω n ] given by F ( β ) := H n − ( B β ( e )) is strictly increasing and smoothly invertible in (0 , nω n ) . (3.5)Therefore, if v : (0 , ∞ ) → [0 , ∞ ) is measurable, thanks to (3.4), there exists a uniquefunction α v : (0 , ∞ ) → [0 , π ] such that ξ v ( r ) = H n − ( B α v ( r ) ( e )) for every r ∈ (0 , ∞ ) . (3.6)Among all the spherically v -distributed sets of R n , we denote by F v the one whose sphericalslices are open geodesic balls centred at the positive e axis., i.e. F v := { x ∈ R n : dist S n − (ˆ x, e ) < α v ( | x | ) } , (3.7)where α v is defined by (3.3) and (3.6). The next result (see [2, Lemma 2.35]) will be usedin the proof of Theorem 1.1. Lemma 3.1.
Let B ⊂ R n be a Borel set and let ϕ h , ϕ : B → R , h ∈ N be summable Borelfunctions such that | ϕ h | ≤ | ϕ | for every h . Then ˆ B sup h ϕ h dx = sup H (cid:40) (cid:88) h ∈ H ˆ A h ϕ h dx (cid:41) , where the supremum ranges over all finite sets H ⊂ N and all finite partitions A h , h ∈ H of B in Borel sets. Normal and tangential components of functions and measures.
For every ϕ ∈ C c ( R n ; R n ), we decompose ϕ as ϕ = ϕ ⊥ + ϕ (cid:107) , where ϕ ⊥ ( x ) := ( ϕ ( x ) · ˆ x ) ˆ x and ϕ (cid:107) ( x ) := ϕ ( x ) − ϕ ⊥ ( x )are the radial and tangential components of ϕ , respectively. If ϕ ∈ C c ( R n ; R n ), div (cid:107) ϕ ( x )stands for the tangential divergence of ϕ at x along the sphere ∂B ( | x | ):div (cid:107) ϕ ( x ) := div ϕ ( x ) − ( ∇ ϕ ( x )ˆ x ) · ˆ x. (3.8)The following lemma gives some useful identities that will be needed later. Lemma 3.2.
Let ϕ ∈ C c ( R n ; R n ) . Then, for every x ∈ R n one has div ϕ ⊥ ( x ) = ( ∇ ϕ ( x )ˆ x ) · ˆ x + ( ϕ ( x ) · ˆ x ) n − | x | , (3.9)div ϕ (cid:107) ( x ) = div (cid:107) ϕ (cid:107) ( x ) . (3.10) Remark 3.3.
Let ϕ ∈ C c ( R n ; R n ) . Recalling that ϕ = ϕ ⊥ + ϕ (cid:107) , combining (3.9) and (3.10) it follows that div ϕ ( x ) = ( ∇ ϕ ( x )ˆ x ) · ˆ x + ( ϕ ( x ) · ˆ x ) n − | x | + div (cid:107) ϕ (cid:107) ( x ) ∀ x ∈ R n . roof. First of all, note that ∇ ( ϕ ( x ) · ˆ x ) = ( ∇ ϕ ( x )) T ˆ x + 1 | x | ϕ (cid:107) ( x ) . (3.11)Indeed, ∇ ( ϕ ( x ) · ˆ x ) = ( ∇ ϕ ( x )) T ˆ x + I − ˆ x ⊗ ˆ x | x | ϕ ( x ) = ( ∇ ϕ ( x )) T ˆ x + 1 | x | ϕ (cid:107) ( x ) , where I represents the identity map in R n , and ˆ x ⊗ ˆ x is the usual tensor product of ˆ x withitself (so that I − ˆ x ⊗ ˆ x is the orthogonal projection on the tangent plane to S n − at ˆ x ).Thanks to (3.11), we havediv ϕ ⊥ ( x ) = div (( ϕ ( x ) · ˆ x )ˆ x ) = ∇ ( ϕ ( x ) · ˆ x ) · ˆ x + ( ϕ ( x ) · ˆ x ) divˆ x = (cid:20) ( ∇ ϕ ( x )) T ˆ x + 1 | x | ϕ (cid:107) ( x ) (cid:21) · ˆ x + ( ϕ ( x ) · ˆ x ) n − | x | = ( ∇ ϕ ( x )ˆ x ) · ˆ x + ( ϕ ( x ) · ˆ x ) n − | x | , which proves (3.9). Note now that, by definition (3.8), it follows thatdiv ϕ ( x ) = div (cid:107) ϕ ( x ) + ( ∇ ϕ ( x )ˆ x ) · ˆ x. (3.12)On the other hand, from (3.9)div ϕ ( x ) = div ϕ (cid:107) ( x ) + div ϕ ⊥ ( x )= div ϕ (cid:107) ( x ) + ( ∇ ϕ ( x )ˆ x ) · ˆ x + ( ϕ ( x ) · ˆ x ) n − | x | . Comparing last identity with (3.12) we obtain that for every ϕ ∈ C c ( R n ; R n )div (cid:107) ϕ ( x ) = div ϕ (cid:107) ( x ) + ( ϕ ( x ) · ˆ x ) n − | x | . Applying the last identity to the function ϕ (cid:107) we obtain (3.10). (cid:3) If µ is an R n -valued Radon measure on R n , we will write µ = µ ⊥ + µ (cid:107) , where µ ⊥ and µ (cid:107) are the R n -valued Radon measures on R n such that ˆ R n ϕ · dµ ⊥ = ˆ R n ϕ ⊥ · dµ, and ˆ R n ϕ · dµ (cid:107) = ˆ R n ϕ (cid:107) · dµ, for every ϕ ∈ C c ( R n ; R n ). Note that µ ⊥ and µ (cid:107) are well defined by Riesz Theorem (see,for instance, [2, Theorem 1.54]). In the special case µ = Df , with f ∈ BV loc ( R n ), we willshorten the notation writing D (cid:107) f and D ⊥ f in place of ( Df ) (cid:107) and ( Df ) ⊥ , respectively.In particular, if f = χ E and E ⊂ R n is a set of finite perimeter, by De Giorgi structuretheorem we have D ⊥ χ E = ν E ⊥ d H n − ∂ ∗ E and D (cid:107) χ E = ν E (cid:107) d H n − ∂ ∗ E. (3.13)Next lemma gives some useful identities concerning the radial and tangential compo-nents of the gradient of a BV loc function. Lemma 3.4.
Let f ∈ BV loc ( R n ) . Then, ˆ R n ϕ ( x ) · dD (cid:107) f = − ˆ R n f ( x ) div (cid:107) ϕ (cid:107) ( x ) dx, (3.14) ˆ R n ϕ ( x ) · dD ⊥ f = − ˆ R n f ( x ) ( ∇ ϕ ( x ) ˆ x ) · ˆ x dx − ˆ R n f ( x ) n − | x | ( ϕ ( x ) · ˆ x ) dx, (3.15) for every ϕ ∈ C c ( R n ; R n ) . roof. Let ϕ ∈ C c ( R n ; R n ). By definition of D (cid:107) f and thanks to (3.10) we have ˆ R n ϕ ( x ) · dD (cid:107) f = ˆ R n ϕ (cid:107) ( x ) · dDf = − ˆ R n div ϕ (cid:107) ( x ) f ( x ) dx = − ˆ R n div (cid:107) ϕ (cid:107) ( x ) f ( x ) dx, and this shows (3.14). Similarly, by definition of D ⊥ f ˆ R n ϕ ( x ) · dD ⊥ f = ˆ R n ϕ ⊥ ( x ) · dDf = − ˆ R n div ϕ ⊥ ( x ) f ( x ) dx. Thanks to (3.9), identity (3.15) follows. (cid:3)
An immediate consequence of identity (3.14) is the following.
Corollary 3.5.
Let f ∈ BV loc ( R n ) and let Ω ⊂⊂ R n be open and bounded. Then, (cid:12)(cid:12) D (cid:107) f (cid:12)(cid:12) (Ω) = sup (cid:26) ˆ R n f ( x ) div (cid:107) ϕ (cid:107) ( x ) dx : ϕ ∈ C c (Ω; R n ) , (cid:107) ϕ (cid:107) L ∞ (Ω; R n ) ≤ (cid:27) . We conclude this subsection with an important proposition, that is a special case of theCoarea Formula (see [2, Theorem 2.93]).
Proposition 3.6.
Let E be a set of finite perimeter in R n and let g : R n → [0 , ∞ ] be aBorel function. Then, ˆ ∂ ∗ E g ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) = ˆ ∞ dr ˆ ( ∂ ∗ E ) r g ( x ) d H n − ( x ) . Proof.
The result follows by applying [2, Remark 2.94] with N = n − M = n , k = 1,and f ( x ) = | x | . (cid:3) In the next subsection we show how the notion of set of finite perimeter can be givenin a natural way also for subsets of the sphere S n − (and, more in general, of ∂B ( r ), forany r > Sets of finite perimeter on S n − . We now give a very brief introduction to sets offinite perimeter on S n − , by using the notion of integer multiplicity rectifiable currents, see[30, Chapter 6] for more details (see also [7]). Let k ∈ N with 1 ≤ k ≤ n −
1. We denoteby Λ k ( R n ) and Λ k ( R n ) the linear spaces of k -vectors and k -covectors in R n , respectively,while D k ( R n ) stands for the set of smooth k -forms with compact support in R n .A k -dimensional current in R n is a continuous linear functional on D k ( R n ). The familyof k -dimensional currents in R n is denoted by D k ( R n ). We say that T ∈ D k ( R n ) is an integer multiplicity rectifiable k -current if it can be represented as T ( ω ) = ˆ M (cid:104) ω ( x ) , η ( x ) (cid:105) θ ( x ) d H k ( x ) for every ω ∈ D k ( R n ) , where M is an H k -measurable countably k -rectifiable subset of R n , θ is an H k -measurablepositive integer-valued function, and η : M → Λ k ( R n ) is an H k -measurable function suchthat for H k -a.e. x ∈ M one has η ( x ) = τ ( x ) ∧ . . . ∧ τ k ( x ), with τ ( x ) , . . . , τ k ( x ) anorthonormal basis for the approximate tangent space of M at x , and (cid:104)· , ·(cid:105) denotes theusual pairing between Λ k ( R n ) and Λ k ( R n ). In the special case when T ( ω ) = ˆ M (cid:104) ω ( x ) , η ( x ) (cid:105) d H k ( x ) for every ω ∈ D k ( R n ) , we write T = [[ M ]]. The boundary ∂T of T is then defined as the element of D k − ( R n )such that ∂T ( ω ) = T ( dω ) for every ω ∈ D k ( R n ) , hile the mass M ( T ) of T is given by M ( T ) := sup (cid:110) T ( ω ) : ω ∈ D k ( R n ) , | ω | ≤ (cid:111) . More in general, for any open set U ⊂ R n , we set M U ( T ) := sup (cid:110) T ( ω ) : ω ∈ D k ( R n ) , | ω | ≤ , supp ω ∈ U (cid:111) . Let A ⊂ S n − be an H n − -measurable set. We will say that A is a set of finite perimeteron S n − if there exists Q ∈ D n − ( R n ) with supp Q ⊂ S n − and Q = ∂ [[ A ]] , with the property that M U ( Q ) < ∞ for every U ⊂⊂ R n . By the Riesz representationtheorem it follows that there exists a Radon measure µ Q and a µ Q -measurable function ν : S n − → T x S n − such that | ν ( x ) | = 1 for µ T -a.e. x and ˆ A div (cid:107) ϕ ( x ) d H n − ( x ) = ˆ S n − ϕ ( x ) · ν ( x ) dµ Q ( x ) , for every smooth vector field with ϕ = ϕ (cid:107) . If A ⊂ S n − is a set of finite perimeter on thesphere, the reduced boundary ∂ ∗ A is the set of points x ∈ S n − such that the limit ν A ( x ) := lim ρ → µ Q ( B ( x, ρ )) ˆ B ( x,ρ ) ν ( y ) dµ Q ( y )exists, ν A ( x ) ∈ T x S n − , and ν A ( x ) = 1. The De Giorgi structure theorem holds true alsofor sets of finite perimeter on the sphere. In particular, ∂ ∗ A is countably ( n − µ Q = H n − ∂ ∗ A , and ˆ A div (cid:107) ϕ ( x ) d H n − ( x ) = ˆ ∂ ∗ A ϕ ( x ) · ν A ( x ) d H n − ( x ) , (3.16)for every smooth vector field with ϕ = ϕ (cid:107) . The isoperimetric inequality on the spherestates that, if β ∈ (0 , π ) and A ⊂ S n − is a set of finite perimeter on S n − with H n − ( A ) = H n − ( B β ( e )), then (see [28]) H n − ( ∂ ∗ B β ( e )) ≤ H n − ( ∂ ∗ A ) . (3.17)The next theorem is a version of a result by Vol’pert (see [32]). Theorem 3.7.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) , and let E ⊂ R n be a spherically v -distributed set of finite perimeter and finite volume. Then, thereexists a Borel set G E ⊂ { α v > } with H ( { α v > } \ G E ) = 0 , such that (i) for every r ∈ G E : (ia) E r is a set of finite perimeter in ∂B ( r ) ; (ib) H n − ( ∂ ∗ ( E r )∆( ∂ ∗ E ) r ) = 0 ; (ii) for every r ∈ G E ∩ { < α v < π } : (iia) | ν E (cid:107) ( rω ) | > , (iib) ν E (cid:107) ( rω ) = ν E r ( rω ) | ν E (cid:107) ( rω ) | ,for H n − -a.e. ω ∈ S n − such that rω ∈ ∂ ∗ ( E r ) ∩ ( ∂ ∗ E ) r .Proof. The result follows applying [30, Theorem 28.5] with f ( x ) = | x | , and recalling thedefinition of slicing of a current (see [30, Definition 28.4]). (cid:3) We now make some important remarks about Theorem 3.7. emark 3.8. Thanks to property (ib), we have ∂ ∗ ( E r ) = H n − ( ∂ ∗ E ) r for every r ∈ G E . Therefore, whenever r ∈ G E we will often write ∂ ∗ E r instead of ∂ ∗ ( E r ) or ( ∂ ∗ E ) r , withoutany risk of ambiguity. Moreover, for every r ∈ G E we will also use the notation p E ( r ) := H n − ( ∂ ∗ E r ) . Remark 3.9.
In dimension n = 2 , the theorem above implies that, if r ∈ G E ∩{ < θ < π } ,then ∂ ∗ ( E r ) = ( ∂ ∗ E ) r and | ν E (cid:107) ( rω ) | > for every ω ∈ S such that rω ∈ ( ∂ ∗ E ) r . (3.18) Let now λ E be the measure defined in (1.5) : λ E ( B ) = ˆ ∂ ∗ E ∩ Φ( B × S ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H ( x ) for every Borel set B ⊂ (0 , ∞ ) . If B ⊂ G E , then by (3.18) | λ E ( B ) | ≤ H ( ∂ ∗ E ∩ Φ( G E × S ) ∩ { ν E (cid:107) = 0 } ) = 0 , so that λ E ( B ) = 0 . As a consequence, λ E is singular with respect to the Lebesgue measurein (0 , ∞ ) . If n > this conclusion is in general false (unless one chooses E = F v , seeRemark 3.10 below), and it may happen that λ E has a non trivial absolutely continuouspart. Remark 3.10. If n ≥ , but we consider the special case E = F v , Theorem 3.7 gives muchmore information than the one we can obtain for a generic set of finite perimeter. Indeed,let R ∈ O ( n ) be any orthogonal transformation that keeps fixed the e axis. By definitionof F v , and thanks to [22, Exercise 15.10] , we have that if x ∈ ∂ ∗ F v , then Rx ∈ ∂ ∗ F v and ν F v (cid:107) ( Rx ) = R ν F v (cid:107) ( x ) and ν F v ⊥ ( Rx ) = R ν F v ⊥ ( x ) . Therefore, applying Theorem 3.7 to F v we infer that (j) for every r ∈ G F v : (ja) ( F v ) r is a spherical cap; (jb) ∂ ∗ ( F v ) r = ( ∂ ∗ F v ) r ; (jj) for every r ∈ G F v ∩ { < α v < π } : (jja) | ν F v (cid:107) ( rω ) | > , (jjb) ν F v (cid:107) ( rω ) = ν ( F v ) r ( rω ) | ν F v (cid:107) ( rω ) | , for every ω ∈ S n − such that rω ∈ ( ∂ ∗ F v ) r ∩ ∂ ∗ ( F v ) r .Therefore, H ( B ) = 0 , (3.19) where B := (cid:110) r ∈ (0 , + ∞ ) : ∃ ω ∈ S n − such that rω ∈ ∂ ∗ F v and ν F v (cid:107) ( rω ) = 0 (cid:111) . Moreover, repeating the argument used in Remark 3.9 one obtains that H n − ( ∂ ∗ F v ∩ Φ( G F v × S n − ) ∩ { ν F v (cid:107) = 0 } ) = 0 . Thus, the measure λ F v defined in (1.5) is purely singular with respect to the Lebesguemeasure in (0 , ∞ ) . . Properties of v and ξ v In this section we discuss several properties of the functions v and ξ v . These are thenatural counterpart in the spherical setting of analogous results proven in [12] and [3]. Westart by showing that, if E ⊂ R n is a set of finite perimeter and volume, then v ∈ BV (0 , ∞ ). Lemma 4.1.
Let v be as in Theorem 1.1, and let E ⊂ R n be a spherically v -distributed setof finite perimeter and finite volume. Then, v ∈ BV (0 , ∞ ) . Moreover, ξ v ∈ BV loc (0 , ∞ ) and ˆ ∞ ψ ( r ) r n − dDξ v ( r ) = ˆ R n ψ ( | x | ) ˆ x · dD ⊥ χ E ( x ) , (4.1) for every bounded Borel function ψ : (0 , ∞ ) → R . As a consequence, | r n − Dξ v | ( B ) ≤ | D ⊥ χ E | (Φ( B × S n − )) , (4.2) for every Borel set B ⊂ (0 , ∞ ) . In particular, r n − Dξ v is a bounded Radon measure on (0 , ∞ ) .Proof. We divide the proof into steps.
Step 1:
We show that v ∈ BV (0 , ∞ ). First of all, note that v ∈ L (0 , ∞ ), since (cid:107) v (cid:107) L (0 , ∞ ) = ˆ ∞ v ( r ) dr = ˆ ∞ dr ˆ ∂B ( r ) χ E ( x ) d H n − ( x ) = H n ( E ) < ∞ . Let now ψ ∈ C c (0 , ∞ ) with | ψ | ≤
1. Applying formula (3.9) to the radial function ψ ( | x | )ˆ x ,we obtain that for every x ∈ R n div ( ψ ( | x | )ˆ x ) = [ ∇ ( ψ ( | x | )ˆ x ) ˆ x ] · ˆ x + [ ψ ( | x | )ˆ x · ˆ x ] n − | x | = (cid:20)(cid:18) ψ (cid:48) ( | x | )ˆ x ⊗ ˆ x + ψ ( | x | ) I − ˆ x ⊗ ˆ x | x | (cid:19) ˆ x (cid:21) · ˆ x + ψ ( | x | ) n − | x | = ψ (cid:48) ( | x | ) + ψ ( | x | ) n − | x | . (4.3)Thus, ˆ R n (cid:20) ψ (cid:48) ( | x | ) + ψ ( | x | ) n − | x | (cid:21) χ E ( x ) dx = ˆ R n div ( ψ ( | x | ) ˆ x ) χ E ( x ) dx = − ˆ R n ψ ( | x | ) ˆ x · dDχ E ( x ) = − ˆ R n ψ ( | x | ) ˆ x · dD ⊥ χ E ( x ) , so that ˆ R n ψ (cid:48) ( | x | ) χ E ( x ) dx (4.4)= − ˆ R n ψ ( | x | ) n − | x | χ E ( x ) dx − ˆ R n ψ ( | x | ) ˆ x · dD ⊥ χ E ( x ) . By Coarea formula, the integral in the left hand side can be written as ˆ R n ψ (cid:48) ( | x | ) χ E ( x ) dx = ˆ ∞ dr ψ (cid:48) ( r ) ˆ ∂B ( r ) χ E ( x ) d H n − ( x ) = ˆ ∞ ψ (cid:48) ( r ) v ( r ) dr. (4.5) ombining (4.4) and (4.5) we find that ˆ ∞ ψ ( r ) dDv ( r )= ˆ R n ψ ( | x | ) n − | x | χ E ( x ) dx + ˆ R n ψ ( | x | ) ˆ x · dD ⊥ χ E ( x ) . (4.6) ≤ ˆ B (1) ψ ( | x | ) n − | x | χ E ( x ) dx + ˆ R n \ B (1) ψ ( | x | ) n − | x | χ E ( x ) dx + P ( E ) ≤ n ( n − ω n ˆ ρ n − dρ + ( n − | E | + P ( E )= nω n + ( n − | E | + P ( E ) < ∞ . Taking the supremum over ψ we obtain that | Dv | (0 , ∞ ) < ∞ , so that v ∈ BV (0 , ∞ ). Step 2:
We conclude the proof. Since the function r (cid:55)→ / ( r n − ) is smooth and locallybounded in (0 , ∞ ), we also have that ξ v ( r ) ∈ BV loc (0 , ∞ ). Moreover, recalling that v ( r ) = r n − ξ v ( r ), by the chain rule in BV (see [2, Example 3.97]) Dv = ( n − r n − ξ v ( r ) dr + r n − Dξ v = ( n − v ( r ) r dr + r n − Dξ v . (4.7)Let now ψ ∈ C c (0 , ∞ ). From the previous identity it follows that ˆ ∞ ψ ( r ) dDv ( r ) = ˆ ∞ ψ ( r ) n − r v ( r ) dr + ˆ ∞ ψ ( r ) r n − dDξ v ( r )= ˆ ∞ ψ ( r ) n − r H n − ( ∂B ( r ) ∩ E ) dr + ˆ ∞ ψ ( r ) r n − dDξ v ( r )= ˆ R n ψ ( | x | ) n − | x | χ E ( x ) dx + ˆ ∞ ψ ( r ) r n − dDξ v ( r ) . Combining the previous identity and (4.6), ˆ ∞ ψ ( r ) r n − dDξ v ( r ) = ˆ R n ψ ( | x | ) ˆ x · dD ⊥ χ E , for every ψ ∈ C c (0 ∞ ) . By approximation, the identity above is true also when ψ is a bounded Borel function,and this gives (4.1).If B ⊂ (0 , ∞ ) is open, thanks to (4.1) we have that for every ψ ∈ C c ( B ) with | ψ | ≤ ˆ B ψ ( r ) r n − dDξ v ( r ) = ˆ Φ( B × S n − ) ψ ( | x | ) ˆ x · dD ⊥ χ E ≤ | D ⊥ χ E | (Φ( B × S n − )) . Taking the supremum over all such ψ gives | r n − Dξ v | ( B ) ≤ | D ⊥ χ E | (Φ( B × S n − )) for every open set B ⊂ (0 , ∞ ) . By approximation, the inequality above holds true for every Borel set, and this showsinequality (4.2). (cid:3)
The next lemma gives an important property of the measure r n − Dξ v . emma 4.2. Let v be as in Theorem 1.1, and let E ⊂ R n be a spherically v -distributedset of finite perimeter and finite volume. Then ( r n − Dξ v )( B ) = ˆ ∂ ∗ E ∩ Φ( B × S n − ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H n − ( x ) (4.8)+ ˆ B dr ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) . for every Borel set B ⊂ (0 , + ∞ ) .Moreover, r n − Dξ v G F v = r n − ξ (cid:48) v dr and for H -a.e. r ∈ G F v ∩ { < α v < π } r n − ξ (cid:48) v ( r ) = H n − ( S α v ( r ) ( re )) ˆ x · ν F v ( x ) | ν F v (cid:107) ( x ) | , for every x ∈ S α v ( r ) ( re ) . Proof.
Let B ⊂ (0 , + ∞ ) be a Borel set. Then, choosing ψ = χ B in (4.1), and recalling(3.13),( r n − Dξ v )( B ) = ˆ + ∞ χ B ( r ) r n − dDξ v ( r )= ˆ Φ( B × S n − ) ˆ x · dD ⊥ χ E ( x ) = ˆ ∂ ∗ E ∩ Φ( B × S n − ) ˆ x · ν E ( x ) d H n − ( x )= ˆ ∂ ∗ E ∩ Φ( B × S n − ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H n − ( x ) + ˆ ∂ ∗ E ∩ Φ( B × S n − ) ∩{ ν E (cid:107) (cid:54) =0 } ˆ x · ν E ( x ) d H n − ( x )= ˆ ∂ ∗ E ∩ Φ( B × S n − ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H n − ( x ) + ˆ B dr ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) , where in the last equality we have used the Coarea formula.Let us now prove the second part of the statement. If one chooses E = F v , thanks toRemark 3.10 we have r n − Dξ v G F v = (cid:32) ˆ ( ∂ ∗ F v ) r ∩{ ν Fv (cid:107) (cid:54) =0 } ˆ x · ν F v ( x ) | ν F v (cid:107) ( x ) | d H n − ( x ) (cid:33) dr G F v = H n − ( S α v ( r ) ( re )) ˆ x · ν F v ( x ) | ν F v (cid:107) ( x ) | . In particular, r n − Dξ v G F v = r n − ξ (cid:48) v ( r ) dr G F v . Moreover, since ξ (cid:48) v ( r ) = 0 H -a.e. in { α = 0 } ∪ { α = π } , we obtain that for H -a.e. r ∈ (0 , ∞ ) r n − ξ (cid:48) ( r ) = H n − ( S α v ( r ) ( re )) ˆ x · ν F v ( x ) | ν F v (cid:107) ( x ) | , for every x ∈ S α v ( r ) ( re ) . (cid:3) We now prove an auxiliary inequality that will be useful later.
Proposition 4.3.
Let v be as in Theorem 1.1, and suppose that there exists a spherically v -distributed set E ⊂ R n of finite perimeter and finite volume. Then, F v is a set of finiteperimeter in R n . Moreover, for every Borel set B ⊂ (0 , + ∞ ) P ( F v ; Φ( B × S n − )) ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) ( B ) + (cid:12)(cid:12) D (cid:107) χ F v (cid:12)(cid:12) (Φ( B × S n − )) . (4.9) roof. The proof is based on the arguments of [12, Lemma 3.5] and [3, Lemma 3.3].Thanks to Lemma 4.1, v ∈ BV (0 , ∞ ). Let { v j } j ∈ N ⊂ C c (0 , ∞ ) be a sequence of non-negative functions such that v j → v H -a.e. in (0 , ∞ ) and | Dv j | ∗ (cid:42) | Dv | . For every j ∈ N ,we denote by F v j ⊂ R n the set defined by (3.7), with v j in place of v . Let now Ω ⊂ (0 , ∞ )be open, and let ϕ ∈ C c (Φ(Ω × S n − ); R n ) with (cid:107) ϕ (cid:107) L ∞ (Φ(Ω × S n − ); R n ) ≤
1. Thanks toRemark 3.3, we have ˆ Φ(Ω × S n − ) χ F vj ( x ) div ϕ ( x ) dx = ˆ Φ(Ω × S n − ) χ F vj ( x ) div (cid:107) ϕ (cid:107) ( x ) dx (4.10)+ ˆ Φ(Ω × S n − ) χ F vj ( x ) ( ∇ ϕ ( x ) ˆ x ) · ˆ x dx + ˆ Φ(Ω × S n − ) χ F vj ( x ) n − | x | ( ϕ ( x ) · ˆ x ) dx. In the following, it will be convenient to introduce the function V j : (0 , ∞ ) → R given by V j ( r ) := ˆ B αvj ( r ) ( re ) ϕ ( x ) · ˆ x d H n − ( x ) = r n − ˆ B αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) , where α v j : (0 , r ) → [0 , π ] is defined by (3.6), with v j in place of v . We divide the proofinto several steps. Step 1:
We show that V j is Lipschitz continuous with compact support. Indeed,supp V j ⊂ Λ(supp ϕ ) := { r ∈ (0 , + ∞ ) : (supp ϕ ) ∩ ∂B ( r ) (cid:54) = ∅} . Moreover, for every r , r ∈ (0 , ∞ ), | V j ( r ) − V j ( r ) | ≤ ˆ B αvj ( r ( e ) | r n − ϕ ( r ω ) · ω − r n − ϕ ( r ω ) · ω | d H n − ( ω )+ r n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ˆ B αvj ( r ( e ) ϕ ( r ω ) · ω d H n − ( ω ) − ˆ B αvj ( r ( e ) ϕ ( r ω ) · ω d H n − ( ω ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c | r − r | + r n − ˆ B αvj ( (cid:101) r ( e ) \ B αvj ( (cid:101) r ( e ) | ϕ ( r ω ) · ω | d H n − ( ω ) ≤ c | r − r | + r n − | ξ v j ( r ) − ξ v j ( r ) | ≤ c | r − r | , where we used the fact that ξ v j is compactly supported in (0 , ∞ ) (since v j is), and (cid:101) r and (cid:101) r are such that α v j ( (cid:101) r ) = max { α v j ( r ) , α v j ( r ) } and α v j ( (cid:101) r ) := min { α v j ( r ) , α v j ( r ) } . Step 2:
We show that α v j is H -a.e. differentiable and that V (cid:48) j ( r ) = ( n − r n − ˆ B αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω )+ r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) (4.11)+ r n − ˆ B αvj ( r ) ( e ) ( ∇ ϕ ( rω ) ω ) · ω d H n − ( ω ) , or H -a.e. r >
0. Let us set A j := { < α v j < π } . Since v j ∈ C c (0 , ∞ ), from (3.5) itfollows that α v j ∈ C ( A j ). Moreover, for every r ∈ A j V (cid:48) j ( r ) = ddr (cid:18) r n − ˆ α vj ( r )0 dβ ˆ S β ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) = ( n − r n − ˆ B αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) + r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) + r n − ˆ α vj ( r )0 dβ ˆ S β ( e ) ( ∇ ϕ ( rω ) ω ) · ω d H n − ( ω )= ( n − r n − ˆ B αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) + r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) + r n − ˆ B αvj ( r ) ( e ) ( ∇ ϕ ( rω ) ω ) · ω d H n − ( ω ) . This shows (4.11) whenever r ∈ A j . Note now that V j ( r ) = 0 for every r ∈ Int( { α v j = 0 } ) ,V j ( r ) = r n − ˆ S n − ϕ ( rω ) · ω d H n − ( ω ) for every r ∈ Int( { α v j = π } ) , where Int( · ) stands for the interior of a set. Since α (cid:48) v j ( r ) = 0 for every r ∈ Int( { α v j =0 } ) ∪ Int( { α v j = π } ), using the identities above one can see that (4.11) holds true for H -a.e. r > Step 3:
We show that ˆ Φ(Ω × S n − ) χ F vj ( x ) ( ∇ ϕ ( x ) ˆ x ) · ˆ x dx + ˆ Φ(Ω × S n − ) χ F vj ( x ) n − | x | ( ϕ ( x ) · ˆ x ) dx = − ˆ Ω dr r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) . Integrating (4.11), thanks to the classical divergence theorem applied in Ω, and recallingthat V j has compact support, we obtain0 = ( n − ˆ Ω dr r n − ˆ B αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω )+ ˆ Ω dr r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) + ˆ Ω dr r n − ˆ B αvj ( r ) ( e ) ( ∇ ϕ ( rω ) ω ) · ω d H n − ( ω )= ˆ Φ(Ω × S n − ) χ F vj ( x ) n − | x | ( ϕ ( x ) · ˆ x ) dx + ˆ Ω dr r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) + ˆ Φ(Ω × S n − ) χ F vj ( x ) ( ∇ ϕ ( x ) ˆ x ) · ˆ x dx, which gives the claim. tep 4: we prove that ˆ Φ(Ω × S n − ) χ F vj ( x ) div ϕ ( x ) dx ≤ (cid:12)(cid:12) r n − Dξ v j (cid:12)(cid:12) (Λ(supp ϕ )) + ˆ Ω H n − ( S α vj ( r ) ) dr, (4.12)where Λ(supp ϕ ) ⊂ (0 , ∞ ) is the compact set defined in Step 1. Thanks to (4.10) andStep 3 ˆ Φ(Ω × S n − ) χ F vj ( x ) div ϕ ( x ) dx = ˆ Φ(Ω × S n − ) χ F vj ( x ) div (cid:107) ϕ (cid:107) ( x ) dx − ˆ Ω dr r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) . (4.13)We now estimate the right hand side of the expression above. Thanks to (3.6) and arguingas in Step 2 we have that ξ (cid:48) v j ( r ) = α (cid:48) v j ( r ) H n − ( S α vj ( r ) ( e )) for H -a.e. r ∈ (0 , ∞ ) . Therefore, − ˆ Ω dr r n − (cid:18) α (cid:48) v j ( r ) ˆ S αvj ( r ) ( e ) ϕ ( rω ) · ω d H n − ( ω ) (cid:19) ≤ ˆ Λ(supp ϕ ) r n − (cid:12)(cid:12)(cid:12) α (cid:48) v j ( r ) (cid:12)(cid:12)(cid:12) H n − ( S α vj ( r ) ( e )) dr (4.14)= ˆ Λ(supp ϕ ) r n − (cid:12)(cid:12)(cid:12) ξ (cid:48) v j ( r ) (cid:12)(cid:12)(cid:12) dr = (cid:12)(cid:12) r n − Dξ v j (cid:12)(cid:12) (Λ(supp ϕ )) . Let us now focus on the second integral in the right hand side of (4.13). Applying thedivergence theorem (3.16) with A = B α vj ( r ) ( re ), and denoting by ν ∗ ( x ) the exterior unitnormal to S α vj ( r ) ( re ), we have ˆ Φ(Ω × S n − ) χ F vj ( x ) div (cid:107) ϕ (cid:107) ( x ) dx = ˆ Ω dr ˆ B αvj ( r ) ( re ) div (cid:107) ϕ (cid:107) ( x ) d H n − ( x )= ˆ Ω dr ˆ S αvj ( r ) ( re ) ϕ (cid:107) ( x ) · ν ∗ ( x ) d H n − ( x ) ≤ ˆ Ω dr H n − ( S α vj ( r ) ( re )) . (4.15)Combining (4.13), (4.14), and (4.15), we obtain (4.12). Step 5:
We show that F v is a set of finite perimeter. Note that χ F vj → χ F v H n -a.e.in R n , and α v j → α H -a.e. in (0 , ∞ ). Note also that, from our choice of the sequence { v j } j ∈ N and thanks to (4.7), it follows that | r n − Dξ v j | ∗ (cid:42) | r n − Dξ v | as j → ∞ . Therefore, taking the limsup as j → ∞ in (4.12), and using the fact that Λ(supp ϕ ) iscompact, ˆ Φ(Ω × S n − ) χ F v ( x ) div ϕ ( x ) dx = lim sup j →∞ ˆ Φ(Ω × S n − ) χ F vj ( x ) div ϕ ( x ) dx ≤ lim sup j →∞ (cid:12)(cid:12) r n − Dξ v j (cid:12)(cid:12) (Λ(supp ϕ )) + lim sup j →∞ ˆ Ω H n − ( S α vj ( r ) ( re )) dr ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Λ(supp ϕ )) + ˆ Ω H n − ( S α v ( r ) ( re )) dr ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Ω) + ˆ Ω H n − ( ∂ ∗ E r ) dr ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Ω) + P ( E ; Φ(Ω × S n − )) , here we also used the isoperimetric inequality in the sphere (see (3.17)) and the Coareaformula. Taking the supremum of the above inequality over all functions ϕ ∈ C c (Φ(Ω × S n − ); R n ) with (cid:107) ϕ (cid:107) L ∞ (Φ(Ω × S n − ); R n ) ≤
1, we obtain P ( F v ; Φ(Ω × S n − )) ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Ω) + P ( E ; Φ(Ω × S n − )) . Thanks to (4.2) we have P ( F v ; Φ(Ω × S n − )) ≤ P ( E ; P ( F v ; Φ(Ω × S n − ))) < ∞ , since E is a set of finite perimeter by assummption. Since Ω was arbitrary, this shows that F v is a set of locally finite perimeter. Step 6:
We conclude. Let Ω ⊂ (0 , ∞ ) be open, and let ϕ ∈ C c (Φ(Ω × S n − ); R n ) with (cid:107) ϕ (cid:107) L ∞ (Φ(Ω × S n − ); R n ) ≤
1. Combining (4.10), Step 3, and (4.14), we have that for every j ∈ N ˆ Φ(Ω × S n − ) χ F vj ( x ) div ϕ ( x ) dx ≤ (cid:12)(cid:12) r n − Dξ v j (cid:12)(cid:12) (Λ(supp ϕ ))+ ˆ Φ(Ω × S n − ) χ F vj ( x ) div (cid:107) ϕ (cid:107) ( x ) dx. Taking the limsup as j → ∞ and thanks to Corollary 3.5, ˆ Φ(Ω × S n − ) χ F v ( x ) div ϕ ( x ) dx ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Λ(supp ϕ )) + ˆ Φ(Ω × S n − ) χ F v ( x ) div (cid:107) ϕ (cid:107) ( x ) dx ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Λ(supp ϕ )) + | D (cid:107) χ F v | (Φ(Ω × S n − )) , where we also used the fact that Λ(supp ϕ ) is compact.Taking the supremum over all ϕ ∈ C c (Φ(Ω × S n − ); R n ) with (cid:107) ϕ (cid:107) L ∞ (Φ(Ω × S n − ); R n ) ≤ P ( F v ; Φ(Ω × S n − )) ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Ω) + | D (cid:107) χ F v | (Φ(Ω × S n − )) , (4.16)which shows (4.9) when B is an open set. Let now B ⊂ (0 , ∞ ) be a Borel set. From (4.16)it follows that P ( F v ; Φ( B × S n − )) ≤ (cid:12)(cid:12) r n − Dξ v (cid:12)(cid:12) (Ω) + P ( E ; Φ(Ω × S n − )) , for any open set Ω ⊂ (0 , ∞ ) with B ⊂ Ω. Taking the infimum of the above inequality overall open sets Ω ⊂ (0 , ∞ ) with B ⊂ Ω, we obtain inequality (4.9) when B is a Borel set. (cid:3) Proof of Theorem 1.1
In this section we prove Theorem 1.1, and state some important auxiliary results. Theproof of Lemma 1.3 is postponed to Section 6, since it requires some results related to thecircular symmetrisation. We start by proving Theorem 1.1.
Proof of Theorem 1.1.
We will adapt the arguments of the proof of [3, Theorem 1.1]. Let G F v be the set associated with F v given by Theorem 3.7. We start by proving (1.4). We willfirst prove the inequality when B ⊂ (0 , ∞ ) \ G F v , and then in the case B ⊂ G F v . The case ofa general Borel set B ⊂ (0 , ∞ ) then follows by decomposing B as B = ( B \ G F v ) ∪ ( B ∩ G F v ). Step 1:
We prove inequality (1.4) when B ⊂ (0 , ∞ ) \ G F v . First observe that, thanks toProposition 3.6 and (3.13), (cid:12)(cid:12) D (cid:107) χ F v (cid:12)(cid:12) (Φ( B × S n − )) = ˆ ∂ ∗ F v ∩ Φ( B × S n − ) | ν F v (cid:107) ( x ) | d H n − ( x ) = ˆ B H n − (( ∂ ∗ F v ) r ) dr = ˆ B ∩{ <α v } H n − (( ∂ ∗ F v ) r ) dr = ˆ B ∩ ( { <α v }\ G Fv ) H n − (( ∂ ∗ F v ) r ) dr = 0 , (5.1) here we used the fact that B ⊂ (0 , ∞ ) \ G F v and H ( { < α v } \ G F v ) = 0. Therefore,thanks to Proposition 4.3 P ( F v ; Φ( B × S n − )) ≤ r n − | Dξ v | ( B ) + (cid:12)(cid:12) D (cid:107) χ F v (cid:12)(cid:12) (Φ( B × S n − ))= r n − | Dξ v | ( B ) ≤ P ( E ; Φ( B × S n − )) , (5.2)where in the last inequality we used (4.2). Step 2:
We prove inequality (1.4) when B ⊂ G F v . We divide this part of the proof intofurther substeps. Step 2a: we prove that P ( E ; Φ( B × S n − )) ≥ P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B (cid:113) p E ( r ) + g ( r ) dr, (5.3)where g : (0 , ∞ ) → R and p E : (0 , ∞ ) → [0 , ∞ ) are defined as g ( r ) := ˆ ∂ ∗ E ∩ ∂B ( r ) ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) and p E ( r ) := H n − ( ∂ ∗ E ∩ ∂B ( r )) , for H -a.e. r ∈ (0 , ∞ ), respectively. We have P ( E ; Φ( B × S n − ))= P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) (cid:54) = 0 } )= P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ ∂ ∗ E ∩ Φ( B × S n − ) ∩{ ν E (cid:107) (cid:54) =0 } d H n − ( x )= P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B dr ˆ ∂ ∗ E ∩ ∂B ( r ) | ν E (cid:107) ( x ) | d H n − ( x )= P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B dr ˆ ∂ ∗ E ∩ ∂B ( r ) (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | (cid:33) d H n − ( x ) , where in the last equality we used the fact that1 = | ν E ⊥ | + | ν E (cid:107) | = (ˆ x · ν E ) + | ν E (cid:107) | . Defining the function f : R → [0 , ∞ ) as f ( t ) := (cid:112) t , we obtain P ( E ; Φ( B × S n − ))= P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B dr ˆ ∂ ∗ E ∩ ∂B ( r ) f (cid:32) ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | (cid:33) d H n − ( x ) . Observing that f is strictly convex, (5.3) follows applying Jensen’s inequality. Step 2b:
We show that ˆ B (cid:113) p E ( r ) + ( r n − ξ (cid:48) v ( r )) dr ≤ P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B (cid:113) p E ( r ) + g ( r ) dr. (5.4) et H ⊂ N be a finite set, and let { A h } h ∈ H be a finite partition of Borel sets of B . Notethat, for each h ∈ H , we have A h ⊂ B ⊂ G F v . Therefore, thanks to Lemma 4.2, for every h ∈ H we have r n − Dξ v A h = r n − ξ (cid:48) v dr A h and ˆ A h w h r n − ξ (cid:48) v ( r ) dr = ˆ A h w h r n − dDξ v ( r )= ˆ ∂ ∗ E ∩ Φ( A h × S n − ) ∩{ ν E (cid:107) =0 } w h ˆ x · ν E ( x ) d H n − ( x )+ ˆ A h dr ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } w h ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x )= ˆ ∂ ∗ E ∩ Φ( A h × S n − ) ∩{ ν E (cid:107) =0 } w h ˆ x · ν E ( x ) d H n − ( x ) + ˆ A h w h g ( r ) dr. (5.5)We will now use the fact that, by duality, we can write (cid:112) t = sup h ∈ N (cid:26) w h t + (cid:113) − w h (cid:27) for every t ∈ R , (5.6)where { w h } h ∈ N is a countable dense set in ( − , (cid:88) h ∈ H ˆ A h (cid:18) w h r n − ξ (cid:48) v ( r ) + p E ( r ) (cid:113) − w h (cid:19) dr = (cid:88) h ∈ H ˆ ∂ ∗ E ∩ Φ( A h × S n − ) ∩{ ν E (cid:107) =0 } w h ˆ x · ν E ( x ) d H n − ( x )+ (cid:88) h ∈ H ˆ A h (cid:16) w h g ( r ) + p E ( r ) (cid:113) − w h (cid:17) dr ≤ (cid:88) h ∈ H ˆ ∂ ∗ E ∩ Φ( A h × S n − ) ∩{ ν E (cid:107) =0 } | ˆ x · ν E ( x ) | d H n − ( x )+ (cid:88) h ∈ H ˆ A h p E ( r ) (cid:18) w h g ( r ) p E ( r ) + (cid:113) − w h (cid:19) dr ≤ (cid:88) h ∈ H (cid:16) P ( E ; Φ( A h × S n − ) ∩ { ν E (cid:107) = 0 } ) (cid:17) + ˆ A h p E ( r ) (cid:115) g ( r ) p E ( r ) dr = P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B (cid:113) p E ( r ) + g ( r ) dr, where we applied identity (5.6) with t = g ( r ) /p E ( r ), and we also used the fact that p E ( r ) = 0 for H -a.e. r / ∈ { < α v < π } , thanks to Volper’t theorem. Applying Lemma 3.1to the functions ϕ h ( r ) = p E ( r ) (cid:18) w h r n − ξ (cid:48) v ( r ) p E ( r ) + (cid:113) − w h (cid:19) , we obtain (5.4). tep 2c: We conclude the proof of Step 2. In the special case E = F v , thanks to Vol’pertTheorem and Lemma 4.2 we have P ( F v ; Φ( B × S n − )) = H n − ( ∂ ∗ F v ∩ Φ( B × S n − ))= ˆ B ∩{ <α v <π } ˆ ∂ ∗ ( F v ) r | ν F v (cid:107) ( x ) | d H n − ( x ) dr = ˆ B ∩{ <α v <π } ˆ ∂ ∗ ( F v ) r (cid:118)(cid:117)(cid:117)(cid:116) (cid:32) ν F v ( x ) | ν F v (cid:107) ( x ) | (cid:33) d H n − ( x ) dr = ˆ B ∩{ <α v <π } (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr. (5.7)Using the isoperimetric inequality (3.17) together with (5.4) and (5.3) we then have, P ( F v ; Φ( B × S n − )) ≤ ˆ B ∩{ <α v <π } (cid:113) p E ( r ) + ( r n − ξ (cid:48) v ( r )) dr ≤ P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B (cid:113) p E ( r ) + g ( r ) dr ≤ P ( E ; Φ( B × S n − )) , from which we conclude. Step 3:
We conclude the proof of the theorem. Suppose P ( E ) = P ( F v ). Then, inparticular, all the inequalities in Step 2 hold true as equalities. At the end of Step 2c weused the fact that, by the isoperimetric inequality (3.17), we have p F v ( r ) ≤ p E ( r ) for H -a.e. r ∈ { < α v < π } . If the above becomes an equality, this means that for H -a.e. r ∈ { < α v < π } the slice E r is a spherical cap. Finally, the fact that for H -a.e. r ∈ { < α v < π } we have H n − ( ∂ ∗ ( E r )∆( ∂ ∗ E ) r ) = 0follows from Vol’pert Theorem 3.7, and this shows (a).Let us now prove (b). If P ( E ) = P ( F v ), the Jensen’s inequality at the end of Step 2b,for the strictly convex function f ( t ) := (cid:112) t , becomes an equality. This implies that for H -a.e. r ∈ { < α v < π } the function x (cid:55)−→ ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | is H n − -a.e. constant in ∂ ∗ E r . Since, for H n − -a.e. x ∈ ∂ ∗ E r , we have1 = | ν E (cid:107) ( x ) | + (ˆ x · ν E ( x )) , this implies that x (cid:55)−→ (ˆ x · ν E ( x )) | ν E (cid:107) ( x ) | = 1 − | ν E (cid:107) ( x ) | is H n − -a.e. constant in ∂ ∗ E r . Therefore, the two functions x (cid:55)−→ ν E ( x ) · ˆ x and x (cid:55)−→ | ν E (cid:107) | ( x )are constant H n − -a.e. in ( ∂ ∗ E ) r . (cid:3) The previous result allows us to prove a useful proposition (see also [3, Proposition 3.4]). roposition 5.1. Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) suchthat F v is a set of finite perimeter and finite volume, let E be a spherically v -distributedset of finite perimeter, and let f : (0 , ∞ ) → [0 , ∞ ] be a Borel function. Then, ˆ ∂ ∗ E f ( | x | ) d H n − ( x ) ≥ ˆ ∞ f ( r ) (cid:113) p E ( r ) + ( r n − ξ (cid:48) v ( r )) dr + ˆ ∞ f ( r ) r n − d | D s ξ v | ( r ) . (5.8) Moreover, in the special case E = F v , equality holds true.Proof. To prove the proposition it is enough to consider the case in which f = χ B , with B ⊂ (0 , ∞ ) Borel set.First, suppose B ⊂ (0 , ∞ ) \ G F v . Thanks to Lemma 4.2, in this case we have ξ (cid:48) v = 0 in B and | r n − Dξ v | ( B ) = | r n − D s ξ v | ( B ). Then, from (4.2) it follows that ˆ ∂ ∗ E χ B ( | x | ) d H n − ( x ) = P ( E ; Φ( B × S n − )) ≥ | D ⊥ χ E | (Φ( B × S n − )) ≥ | r n − Dξ v | ( B ) = | r n − D s ξ v | ( B ) = ˆ ∞ χ B ( r ) r n − d | D s ξ v | ( r )= ˆ ∞ χ B ( r ) (cid:113) p E ( r ) + ( r n − ξ (cid:48) v ( r )) dr + ˆ ∞ χ B ( r ) r n − d | D s ξ v | ( r ) , where we also used the fact that p E = 0 H -a.e. in B , since H n ( E ∩ Φ( B × S n − )) ≤ ˆ { v =0 } dr ˆ E r d H n − ( x ) = ˆ { v =0 } v ( r ) dr = 0 . Let us now assume B ⊂ G F v . In this case, by Lemma 4.2 we have | r n − D s ξ v | ( B ) = 0.Then, thanks to (5.3) and (5.4) we obtain ˆ ∂ ∗ E χ B ( | x | ) d H n − ( x ) = P ( E ; Φ( B × S n − )) ≥ P ( E ; Φ( B × S n − ) ∩ { ν E (cid:107) = 0 } ) + ˆ B (cid:113) p E ( r ) + g ( r ) dr ≥ ˆ B (cid:113) p E ( r ) + ( r n − ξ (cid:48) v ( r )) dr = ˆ ∞ χ B ( r ) (cid:113) p E ( r ) + ( r n − ξ (cid:48) v ( r )) dr + ˆ ∞ χ B ( r ) r n − d | D s ξ v | ( r ) , so that (5.8) follows.Consider now the case E = F v . If B ⊂ G F v , recalling again that by Lemma 4.2 we have | r n − D s ξ v | ( B ) = 0, thanks to (5.7) we obtain ˆ ∂ ∗ F v χ B ( | x | ) d H n − ( x ) = P ( F v ; Φ( B × S n − )) = ˆ B (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr = ˆ ∞ χ B ( r ) (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr + ˆ ∞ χ B ( r ) r n − d | D s ξ v | ( r ) . If, instead, B ⊂ (0 , ∞ ) \ G F v , then ξ (cid:48) v = 0 in B and | r n − Dξ v | ( B ) = | r n − D s ξ v | ( B ).Therefore, thanks to (5.2), ˆ ∂ ∗ F v χ B ( | x | ) d H n − ( x ) = P ( F v ; Φ( B × S n − )) ≤ r n − | Dξ v | ( B ) = | r n − D s ξ v | ( B )= ˆ ∞ χ B ( r ) (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr + ˆ ∞ χ B ( r ) r n − d | D s ξ v | ( r ) . (cid:3) n important consequence of the above proposition is a formula for the perimeter of F v . Corollary 5.2.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) suchthat F v is a set of finite perimeter and finite volume. Then P ( F v ; Φ( B × S n − )) = ˆ B (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr + ˆ B r n − d | D s ξ v | ( r ) . (5.9)We conclude this section with an important result, that will be used later. Proposition 5.3.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) suchthat F v is a set of finite perimeter and finite volume, and let I ⊂ (0 , + ∞ ) be an open set.Then the following three statements are equivalent: (i) H n − (cid:16)(cid:110) x ∈ ∂ ∗ F v ∩ Φ( I × S n − ) : ν F v (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 ; (ii) ξ v ∈ W , ( I ) ; (iii) P ( F v ; Φ( B × S n − )) = 0 for every Borel set B ⊂ I , such that H ( B ) = 0 . Remark 5.4.
Note that the equivalence (iii) ⇐⇒ (i) holds true also if I is a Borel set. Toshow this, we only need to prove that (i) = ⇒ (iii) , since the opposite implication is givenby repeating Step 3 of the proof of Proposition 5.3. Suppose (i) is satisfied. Then from (4.8) we have r n − Dξ v I = r n − ξ (cid:48) v I . Therefore, thanks to (5.9) P ( F v ; Φ( B × S n − )) = ˆ B (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr for every Borel set B ⊂ I, which implies (iii).Proof. We divide the proof into three steps.
Step 1: (i) = ⇒ (ii). Recall that, by Lemma 4.1, ξ v ∈ BV loc ( I ). If (i) is satisfied, from(4.8) we have r n − Dξ v I = r n − ξ (cid:48) v I , which implies (ii). Step 2: (ii) = ⇒ (iii). This implication follows from formula (5.9). Step 3: (iii) = ⇒ (i) (note that we will not use the fact that I is open). Assume (iii) holdstrue. Then, H n − (cid:16)(cid:110) x ∈ ∂ ∗ F v ∩ Φ( I × S n − ) : ν ∂ ∗ F v (cid:107) ( x ) = 0 (cid:111)(cid:17) ≤ P ( ∂ ∗ F v ; Φ(( B ∩ I ) × S n − )) = 0 , where we used the fact that H ( B ) = 0, thanks to (3.19). (cid:3) Circular symmetrisation and proof of Lemma 1.3
In this section we show Theorem 1.4, Lemma 1.5, and finally Lemma 1.3. We willonly sketch the proofs, since in most cases the arguments follow the lines of the proofs inSection 3, Section 4, and Section 5.We start with some notation which, together with that one already given in the Intro-duction, will be extensively used in this section. Let ( r, x (cid:48) ) ∈ (0 , ∞ ) × R n − , β ∈ [0 , π ],and let p ∈ S . The circular arc of centre ( rp, x (cid:48) ) and radius β is the set B β ( rp, x (cid:48) ) := { x ∈ ∂B ((0 , x (cid:48) ) , r ) ∩ Π x (cid:48) : dist S (ˆ x , rp ) < β } , If (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) is a measurable function satisfying (1.10), we define α (cid:96) :(0 , ∞ ) × R n − → [0 , π ] and ξ (cid:96) : (0 , ∞ ) × R n − → [0 , π ] as α (cid:96) := 12 r (cid:96) ( r, x (cid:48) ) and ξ (cid:96) ( r, x (cid:48) ) = 1 r (cid:96) ( r, x (cid:48) ) = 2 α (cid:96) ( r, x (cid:48) ) . ote that in this case the relation between α (cid:96) and ξ (cid:96) is linear. If µ is an R n -valued Radonmeasure on R n \ { x = 0 } , we will write µ = µ ⊥ + µ (cid:107) , where µ ⊥ and µ (cid:107) are the R n -valued Radon measures on R n \ { x = 0 } such that ˆ R n \{ x =0 } ϕ · dµ ⊥ = ˆ R n \{ x =0 } ϕ ⊥ · dµ, and ˆ R n \{ x =0 } ϕ · dµ (cid:107) = ˆ R n \{ x =0 } ϕ (cid:107) · dµ, for every ϕ ∈ C c ( R n \ { x = 0 } ; R n ). The next two results play the role of Proposition 3.6and Vol’pert Theorem 3.7, in the context of circular symmetrisation. Proposition 6.1.
Let E be a set of finite perimeter in R n and let g : R n → [0 , ∞ ] be aBorel function. Then, ˆ ∂ ∗ E g ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) = ˆ (0 , ∞ ) × R n − dr dx (cid:48) ˆ ( ∂ ∗ E ) ( r,x (cid:48) ) g ( x ) d H ( x ) . Proof.
In this case, the result follows applying [2, Remark 2.94] with N = n − M = n , k = n −
1, and f ( x ) = ( | x | , x (cid:48) ). (cid:3) Theorem 6.2.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) ,and let E ⊂ R n be an circularly (cid:96) -distributed set of finite perimeter and finite volume.Then, there exists a Borel set G (cid:96)E ⊂ { α (cid:96) > } with H n − ( { α (cid:96) > } \ G (cid:96)E ) = 0 , such that (i) for every ( r, x (cid:48) ) ∈ G (cid:96)E : (ia) E ( r,x (cid:48) ) is a set of finite perimeter in ∂B r (0 , x (cid:48) ) ∩ Π x (cid:48) ; (ib) ∂ ∗ ( E ( r,x (cid:48) ) ) = ( ∂ ∗ E ) ( r,x (cid:48) ) ; (ii) for every ( r, x (cid:48) ) ∈ G (cid:96)E ∩ { < α (cid:96) < π } : (iia) | ν E (cid:107) ( rω, x (cid:48) ) | > ; (iib) ν E (cid:107) ( rω, x (cid:48) ) = ν E ( r,x (cid:48) ) ( rω, x (cid:48) ) | ν E (cid:107) ( rω, x (cid:48) ) | ,for every ω ∈ S such that ( rω, x (cid:48) ) ∈ ∂ ∗ ( E ( r,x (cid:48) ) ) = ( ∂ ∗ E ) ( r,x (cid:48) ) .Proof. The statement follows applying the results of [18, Section 2.5], where the slicing ofcodimension higher than 1 for currents is defined. (cid:3)
Remark 6.3.
Note that, if ( r, x (cid:48) ) ∈ G (cid:96)E , conditions (iia) and (iib) are satisfied for every ω ∈ S such that ( rω, x (cid:48) ) ∈ ∂ ∗ ( E ( r,x (cid:48) ) ) = ( ∂ ∗ E ) ( r,x (cid:48) ) . This is due to the fact that the cir-cular symmetrisation has codimension . Such property fails, in general, for the sphericalsymmetrisation (see Remark 3.9). Remark 6.4.
An argument similar to that one used in Remark 3.9 shows that H n − ( ∂ ∗ E ∩ Φ ( G (cid:96)E × S ) ∩ { ν E (cid:107) = 0 } ) = 0 . As a consequence, the measure λ (cid:96)E defined as: λ (cid:96)E ( B ) := ˆ ∂ ∗ E ∩ Φ ( B × S ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H ( x ) , for every Borel set B ⊂ (0 , ∞ ) × R n − , is singular with respect to the Lebesgue measure in (0 , ∞ ) × R n − . The following result plays the role of Lemma 4.1 in the context of circular symmetrisa-tion. emma 6.5. Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) ,and let E ⊂ R n be an circularly (cid:96) -distributed set of finite perimeter and finite volume.Then, (cid:96) ∈ BV loc ((0 , ∞ ) × R n − ) . Moreover, ξ (cid:96) ∈ BV loc ((0 , ∞ ) × R n − ) and ˆ (0 , ∞ ) × R n − ψ ( r, x (cid:48) ) r dD r ξ (cid:96) ( r, x (cid:48) ) = ˆ R n \{ x =0 } ψ ( | x | , x (cid:48) ) ˆ x · dD ⊥ χ E ( x ) , for every bounded Borel function ψ : (0 , ∞ ) × R n − → R , where D r ξ (cid:96) denotes the r -component of the R n − -valued Radon measure Dξ (cid:96) . As a consequence, | rD r ξ (cid:96) | ( B ) ≤ | D ⊥ χ E | (Φ ( B × S )) , for every Borel set B ⊂ (0 , ∞ ) × R n − . In particular, rD r ξ (cid:96) is a bounded Radon measureon (0 , ∞ ) × R n − . Finally, D x (cid:48) (cid:96) ( B ) = ˆ ∂ ∗ E ∩ Φ ( B × S ) ν Ex (cid:48) ( x ) d H n − ( x ) , for every Borel set B ⊂ (0 , ∞ ) × R n − . Remark 6.6.
Unlike what happened when we were considering the spherical symmetrisa-tion, now the function (cid:96) might fail to be in BV ((0 , ∞ ) × R n − ) . Indeed, in Step 1 of theproof of Lemma 4.1 we used the fact that for r bounded we are in a bounded set. This isnot true in the context of circular symmetrisation. The next lemma, which is related to Lemma 4.2, will show the advantage of consideringa symmetrisation of codimension 1.
Lemma 6.7.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) ,and let E ⊂ R n be an circularly (cid:96) -distributed set of finite perimeter and finite volume.Then ( r dD r ξ (cid:96) )( B ) = ˆ ∂ ∗ E ∩ Φ ( B × S ) ∩{ ν E (cid:107) =0 } ˆ x · ν E ( x ) d H n − ( x )+ ˆ B dr dx (cid:48) ˆ ( ∂ ∗ E ) ( r,x (cid:48) ) ∩{ ν E (cid:107) (cid:54) =0 } ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H ( x ) . for every Borel set B ⊂ (0 , ∞ ) × R n − . Moreover, r ( ξ (cid:96) ) (cid:48) ( r, x (cid:48) ) = ˆ ( ∂ ∗ E ) ( r,x (cid:48) ) ∩{ ν E (cid:107) (cid:54) =0 } ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H ( x ) , for H n − -a.e. ( r, x (cid:48) ) ∈ (0 , ∞ ) × R n − , where ( ξ (cid:96) ) (cid:48) denotes the approximate differential of ξ (cid:96) with respect to r . Similarly, D x (cid:48) (cid:96) ( B ) = ˆ ∂ ∗ E ∩ Φ ( B × S ) ∩{ ν E (cid:107) =0 } ν Ex (cid:48) ( x ) d H n − ( x )+ ˆ B dr dx (cid:48) ˆ ( ∂ ∗ E ) ( r,x (cid:48) ) ∩{ ν E (cid:107) (cid:54) =0 } ν Ex (cid:48) ( x ) | ν E (cid:107) ( x ) | d H ( x ) . for every Borel set B ⊂ (0 , ∞ ) × R n − , and ∇ x (cid:48) (cid:96) ( r, x (cid:48) ) = ˆ ( ∂ ∗ E ) ( r,x (cid:48) ) ∩{ ν E (cid:107) (cid:54) =0 } ν Ex (cid:48) ( x ) | ν E (cid:107) ( x ) | d H ( x ) , for H n − -a.e. ( r, x (cid:48) ) ∈ (0 , ∞ ) × R n − , where ∇ x (cid:48) (cid:96) denotes the approximate gradient of (cid:96) with respect to x (cid:48) . The next result should be compared to Proposition 4.3. roposition 6.8. Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) , and suppose that there exists an circularly (cid:96) -distributed set E ⊂ R n be of finiteperimeter and finite volume. Then, F (cid:96) is a set of finite perimeter in R n . Moreover, forevery Borel set B ⊂ (0 , + ∞ ) × R n − P ( F (cid:96) ; Φ ( B × S )) ≤ | D x (cid:48) (cid:96) | ( B ) + | rD r ξ (cid:96) | ( B ) + (cid:12)(cid:12) D (cid:107) χ F v (cid:12)(cid:12) (Φ ( B × S )) . We are now ready to prove Theorem 1.4.
Proof of Theorem 1.4.
Using the results shown above, Theorem 1.4 can be proved by fol-lowing the lines of the proof of Theorem 1.1. (cid:3)
We will now state the results that are need to prove Lemma 1.5. The next propositionshould be compared to Proposition 5.1.
Proposition 6.9.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) such that F (cid:96) is a set of finite perimeter and finite volume, let E ⊂ R n be an circularly (cid:96) -distributed set of finite perimeter, and let f : (0 , ∞ ) × R n − → [0 , ∞ ] be a Borel function.Then, ˆ ∂ ∗ E f ( | x | , x (cid:48) ) d H n − ( x ) ≥ ˆ (0 , ∞ ) × R n − f ( r, x (cid:48) ) (cid:113) p E ( r, x (cid:48) ) + ( r ( ξ (cid:96) ) (cid:48) ( r, x (cid:48) )) + |∇ x (cid:48) (cid:96) ( r, x (cid:48) ) | dr dx (cid:48) + ˆ (0 , ∞ ) × R n − f ( r, x (cid:48) ) r d | D sr ξ (cid:96) | ( r, x (cid:48) ) + ˆ (0 , ∞ ) × R n − f ( r, x (cid:48) ) d | D sx (cid:48) (cid:96) | ( r, x (cid:48) ) . Moreover, in the special case E = F (cid:96) , equality holds true. A straightforward consequence of the previous result is the following formula for theperimeter of F (cid:96) . Corollary 6.10.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) such that F (cid:96) is a set of finite perimeter and finite volume. Then P ( F (cid:96) ; Φ ( B × S ))= ˆ B (cid:113) p E ( r, x (cid:48) ) + ( r ( ξ (cid:96) ) (cid:48) ( r, x (cid:48) )) + |∇ x (cid:48) (cid:96) ( r, x (cid:48) ) | dr dx (cid:48) + | rD sr ξ (cid:96) | ( B ) + | D sx (cid:48) (cid:96) | ( B ) . Next lemma relies on the fact that the circular symmetrisation has codimension 1. Theproof can be obtained by repeating the arguments used in the proof of [12, Lemma 4.1].
Lemma 6.11.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) ,let E ⊂ R n be an circularly (cid:96) -distributed set of finite perimeter and finite volume, and let A ⊂ (0 , + ∞ ) × R n − be a Borel set. Then, H n − (cid:16) { x ∈ ∂ ∗ E : ν E (cid:107) ( x ) = 0 } ∩ Φ ( A × S ) (cid:17) = 0 . if and only if P ( E ; Φ ( B × S )) = 0 for every Borel set B ⊂ A with H n − ( B ) = 0 . The next proposition can be proved with the same arguments used to show Proposi-tion 5.3.
Proposition 6.12.
Let (cid:96) : (0 , ∞ ) × R n − → [0 , ∞ ) be a measurable function satisfying (1.10) such that F (cid:96) is a set of finite perimeter and finite volume, and let Ω ⊂ (0 , + ∞ ) × R n − be an open set. Then the following three statements are equivalent: i) H n − (cid:16)(cid:110) x ∈ ∂ ∗ F (cid:96) ∩ Φ (Ω × S ) : ν F (cid:96) (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 ; (ii) ξ (cid:96) ∈ W , (Ω) and (cid:96) ∈ W , (Ω) ; (iii) P ( F (cid:96) ; Φ ( B × S )) = 0 for every Borel set B ⊂ Ω , such that H n − ( B ) = 0 .Proof of Lemma 1.5. Once all the results above are established, Lemma 1.5 can be shownby adapting the arguments used in the proof of [12, Proposition 4.2]. (cid:3)
We can now prove Lemma 1.3. As already mentioned in the Introduction, the proofrelies on Theorem 1.4 and Lemma 1.5.
Proof of Lemma 1.3.
We divide the proof into steps.
Step 1:
We show that (1.8) = ⇒ (1.9). Suppose (1.8) is satisfied. Then, from (4.8) wehave r n − Dξ v I = r n − ξ (cid:48) v I . Thanks to (5.9), this implies that P ( F v ; Φ( B × S n − )) = ˆ B (cid:113) p F v ( r ) + ( r n − ξ (cid:48) v ( r )) dr. for every Borel set B ⊂ I. In particular, condition (iii) of Proposition 5.3 is satisfied. Then, (1.9) follows from Re-mark 5.4.
Step 2:
We show that if P ( E ; Φ( I × S n − )) = P ( F v ; Φ( I × S n − )), then (1.9) implies (1.8).To this aim, we first prove an auxiliary result. Step 2a:
We show that if F ⊂ R n is a set of finite perimeter such that ( F ) r is a sphericalcap for H -a.e. r >
0, and H n − (cid:16)(cid:110) x ∈ ∂ ∗ F ∩ Φ( I × S n − ) : ν F (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 , (6.1)then H n − ( B j ) = 0 for every j = 2 , . . . , n , where B j := (cid:110) x ∈ ∂ ∗ F ∩ Φ( I × S n − ) : ν F j (cid:107) ( x ) = 0 (cid:111) . Here, the vector ν F j (cid:107) is defined in the following way. Let j ∈ { , . . . , n } , and let ν F j be theorthogonal projection of ν F on the bi-dimensional plane generated by e and e j . In thisplane, we consider the following orthonormal basis { (cid:98) x j , (cid:101) x j } : (cid:98) x j = 1 (cid:113) x + x j ( x , j − (cid:122) (cid:125)(cid:124) (cid:123) , . . . , , x j , n − j times (cid:122) (cid:125)(cid:124) (cid:123) , . . . , , and (cid:101) x j = 1 (cid:113) x + x j ( − x j , j − (cid:122) (cid:125)(cid:124) (cid:123) , . . . , , x , n − j times (cid:122) (cid:125)(cid:124) (cid:123) , . . . , , where (cid:98) x j is directed along the radial direction, and (cid:101) x j is parallel to the tangential direc-tion. To show the claim, first of all note that, by Vol’pert Theorem 3.7, for H -a.e. r > B j ) r = (cid:110) x ∈ ∂ ∗ F r ∩ Φ( I × S n − ) : ν F r (cid:107) ( x ) · (cid:101) x j = 0 (cid:111) . up to an H n − -negligible set. Since ( B j ) r is a spherical cap, we have H n − (( B j ) r ) = 0.Then, thanks to (6.1), H n − ( B j ) = H n − (cid:16) B j ∩ (cid:110) x ∈ ∂ ∗ F ∩ Φ( I × S n − ) : ν F (cid:107) ( x ) (cid:54) = 0 (cid:111)(cid:17) = ˆ I dr ˆ ∂ ∗ F r ∩ ( B j ) r χ { ν F (cid:107) (cid:54) =0 } ( x ) 1 | ν F (cid:107) ( x ) | d H n − ( x ) = 0 . tep 2b: We conclude. Let E := E , and let E be set obtained by applying to E the circular symmetrisation with respect to ( e , e ). Then, for j = 3 , . . . , n , we defineiteratively the set E j as the circular symmetral of E j − with respect to ( e , e j ). Notethat, since H -a.e. spherical section of E is a spherical cap, we have E n = F v . Therefore,thanks to the perimeter inequality (1.11) under circular symmetrisation (see Theorem 1.4),we have P ( F v ; Φ( I × S n − )) = P ( E n − ; Φ( I × S n − )) = . . . = P ( E ; Φ( I × S n − )) . Moreover, for j = 3 , . . . , n , we define I j := Φ( I × S n − ) ∩ { x j = 0 } ∩ { x > } . It is notdifficult to check that Φ( I × S n − ) = Φ j ( I j × S ) for j = 3 , . . . , n. Then, applying Lemma 1.5 to F v and E n − , we obtain that H n − (cid:16)(cid:110) x ∈ ∂ ∗ E n − ∩ Φ n − ( I n − × S ) : ν E n − n − (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 , which, in turns, implies H n − (cid:16)(cid:110) x ∈ ∂ ∗ E n − ∩ Φ n − ( I n − × S ) : ν E n − (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . Applying iteratively this argument to E n − , . . . , E , we conclude. (cid:3) Proof of Theorem 1.2: (ii) = ⇒ (i) Before giving the proof of the implication (ii) = ⇒ (i) of Theorem 1.2, it will be convenientto introduce some useful notation. Let v and I = { < α ∧ v ≤ α ∨ v < π } be as in thestatement of Theorem 1.2. By assumption, I is an interval and α v ∈ W , ( I ) where, toease the notation, we set I := ˚ I . Let now E be a spherically v -distributed set of finiteperimeter. We define the average direction of E as the map d E : I → S n − given by d E ( r ) := ω n − (sin α v ( r )) n − r n − ˆ E r ˆ x d H n − ( x ) , if r ∈ I ∩ G E ,e otherwise in I, (7.1)where G E ⊂ (0 , ∞ ) is the set given by Theorem 3.7. To ease our calculations, it will alsobe convenient to introduce the barycentre function b E : I → R n of E as b E ( r ) := r n − ˆ E r ˆ x d H n − ( x ) , if r ∈ I ∩ G E ,e otherwise in I. The importance of the functions d E and b E is given by the following lemma. Lemma 7.1.
Let v be as in Theorem 1.2, let I ⊂ (0 , ∞ ) be an open interval, and let E be a spherically v -distributed set of finite perimeter such that E r is H n − -equivalent to aspherical cap for H -a.e. r ∈ I . Then, E ∩ Φ( I × S n − ) = H n { x ∈ Φ( I × S n − ) : dist S n − (ˆ x, d E ( | x | )) < α v ( | x | ) } . Moreover, b E ( r ) = ω n − (sin α v ( r )) n − d E ( r ) for H -a.e. r ∈ I. (7.2) Proof.
Let us immediately observe that (7.2) follows by the definitions of d E and b E . Byassumption, for H -a.e. r ∈ I , there exists ω ( r ) ∈ S n − such that E r = B α v ( r ) ( rω ( r )). Weare left to show that ω ( r ) = d E ( r ) for H -a.e. r ∈ I. (7.3) ote that for H -a.e. r ∈ I we have E r = B α v ( r ) ( rω ( r )) and ∂ ∗ E r = S α v ( r ) ( rω ( r )).Therefore, for H -a.e. r ∈ I ˆ E r ˆ x d H n − ( x ) = ˆ α v ( r )0 dβ ˆ S β ( rω ( r )) x d H n − ( x ) . (7.4)Observe now that, thanks to the symmetry of the geodesic sphere and recalling (3.2), forevery β ∈ (0 , α v ( r )) we have ˆ S β ( rω ( r )) x d H n − ( x ) = (cid:32) ˆ S β ( rω ( r )) ( x · ω ( r )) d H n − ( x ) (cid:33) ω ( r ) (7.5)= r cos β H n − ( S β ( rω ( r ))) ω ( r ) = ( n − ω n − r n − cos β (sin β ) n − ω ( r ) . Combining (7.4) and (7.5) we obtain that for H -a.e. r ∈ I ˆ E r ˆ x d H n − ( x ) = ( n − ω n − r n − (cid:32) ˆ α v ( r )0 cos β (sin β ) n − dβ (cid:33) ω ( r )= ω n − r n − (sin α v ( r )) n − ω ( r ) . Recalling the definition of d E , identity (7.3) follows. (cid:3) Remark 7.2.
Let us point out that here we are using the term barycentre in a slightlyimprecise way. Indeed, for a given r ∈ I ∩ G E , the geometric barycentre of E r is given by H n − ( E r ) ˆ E r x d H n − ( x ) = 1 ξ v ( r ) r n − ˆ E r x d H n − ( x )= rξ v ( r ) 1 r n − ˆ E r ˆ x d H n − ( x ) = rξ v ( r ) b E ( r ) . Nevertheless, we will still keep this terminology, since b E turns out to be very useful forour analysis. We are now ready to prove the implication (ii) = ⇒ (i) of Theorem 1.2. Proof of Theorem 1.2: (ii) = ⇒ (i). Suppose (ii) is satisfied, and let E ∈ N ( v ). We aregoing to show that there exists an orthogonal transformation R ∈ SO ( n ) such that H n ( E ∆( RF v )) = 0. We now divide the proof into steps. Step 1:
First of all, we observe that H n − (cid:16)(cid:110) x ∈ ∂ ∗ E ∩ Φ( I × S n − ) : ν E (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . Indeed, since α v ∈ W , ( I ), thanks to Proposition 5.3 we have H n − (cid:16)(cid:110) x ∈ ∂ ∗ F v ∩ Φ( I × S n − ) : ν F v (cid:107) ( x ) = 0 (cid:111)(cid:17) = 0 . Since E ∈ N ( v ), applying Lemma 1.3 the claim follows. Step 2:
We show that b E ∈ W , ( I ; R n ) and b (cid:48) E ( r ) = 1 r n ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } x ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) . (7.6) ndeed, let ψ ∈ C c ( I ) be arbitrary, and let i ∈ { , . . . , n } . By definition of b E ˆ I ( b E ) i ( r ) ψ (cid:48) ( r ) dr = ˆ I ˆ E ∩ ∂B ( r ) r n − x i | x | d H n − ( x ) ψ (cid:48) ( r ) dr = ˆ Φ( I × S n − ) x i | x | n ψ (cid:48) ( | x | ) χ E ( x ) dx. Note now that div (cid:18) x i | x | n ψ ( | x | )ˆ x (cid:19) = x i | x | n ψ (cid:48) ( | x | ) . Indeed, recalling (4.3),div (cid:18) x i | x | n ψ ( | x | )ˆ x (cid:19) = ψ ( | x | ) ∇ (cid:18) x i | x | n (cid:19) · ˆ x + x i | x | n div( ψ ( | x | )ˆ x )= ψ ( | x | ) (cid:18) e i | x | n − n x i | x | n +1 ˆ x (cid:19) · ˆ x + x i | x | n (cid:18) ψ (cid:48) ( | x | ) + ψ ( | x | ) n − | x | (cid:19) = x i | x | n ψ (cid:48) ( | x | ) . Therefore, ˆ I ( b E ) i ( r ) ψ (cid:48) ( r ) dr = ˆ Φ( I × S n − ) div (cid:18) x i | x | n ψ ( | x | )ˆ x (cid:19) χ E ( x ) dx = − ˆ Φ( I × S n − ) x i | x | n ψ ( | x | )ˆ x · dDχ E ( x )= ˆ ∂ ∗ E ∩ Φ( I × S n − ) x i | x | n ψ ( | x | ) ˆ x · ν E ( x ) d H n − ( x ) . Thanks to Step 1 we then obtain ˆ I ( b E ) i ( r ) ψ (cid:48) ( r ) dr = ˆ ∂ ∗ E ∩{ ν E (cid:107) (cid:54) =0 }∩ Φ( I × S n − ) x i | x | n ψ ( | x | ) ˆ x · ν E ( x ) d H n − ( x )= ˆ I ψ ( r ) 1 r n (cid:34) ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } x i ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x ) (cid:35) dr, so that (7.6) follows. Step 3:
We show that b (cid:48) E ( r ) = ( n − α (cid:48) v ( r ) cos α v ( r )sin α v ( r ) b E ( r ) for H -a.e. r ∈ I. (7.7)Since E ∈ N ( v ), from Theorem 1.1 we know that for H -a.e. r ∈ I the spherical slice E r is a spherical cap. Then, thanks to Lemma 7.1 E r = B α v ( r ) ( rd E ( r )) and ( ∂ ∗ E ) r = S α v ( r ) ( rd E ( r )) for H -a.e. r ∈ I. Still thanks to Theorem 1.1, we know that for H -a.e. r ∈ I the functions x (cid:55)→ ν E ( x ) · ˆ x and x (cid:55)→ | ν E (cid:107) | ( x ) are constant H n − -a.e. in ( ∂ ∗ E ) r , say ν E ( x ) · ˆ x = a ( r ) and | ν E (cid:107) | ( x ) = c ( r ) , for H -a.e. r ∈ I, or some measurable functions a : I → ( − ,
1) and c : I → (0 , d E together with (7.4)-(7.5) we obtain b (cid:48) E ( r ) = 1 r n ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } x ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x )= 1 r n a ( r ) c ( r ) ˆ S αv ( r ) ( rd E ( r )) x d H n − ( x )= 1 r n a ( r ) c ( r ) r cos( α v ( r )) H n − ( S α v ( r ) ( rd E ( r ))) d E ( r )= 1 r n − a ( r ) c ( r ) H n − ( S α v ( r ) ( rd E ( r ))) cos( α v ( r )) d E ( r ) . (7.8)Note now that from Step 1 and (4.8) it follows that for H -a.e. r ∈ Ir n − ξ (cid:48) v ( r ) = ˆ ( ∂ ∗ E ) r ∩{ ν E (cid:107) (cid:54) =0 } ˆ x · ν E ( x ) | ν E (cid:107) ( x ) | d H n − ( x )= a ( r ) c ( r ) H n − ( S α v ( r ) ( rd E ( r ))) . Plugging last identity into (7.8) and using (7.2), we obtain b (cid:48) E ( r ) = ξ (cid:48) v ( r ) cos( α v ( r )) d E ( r ) = ξ (cid:48) v ( r ) cos( α v ( r )) b E ( r ) ω n − (sin α v ( r )) n − = ( n − α (cid:48) v ( r ) cos α v ( r )sin α v ( r ) b E ( r ) , where we used the fact that, thanks to (3.1) and (3.3), ξ (cid:48) v ( r ) = ( n − ω n − (sin α v ( r )) n − α (cid:48) v ( r ) for H -a.e. r ∈ I. Step 4:
We conclude. First of all, note that from (7.2) and Step 2 it follows that d E ∈ W , ( I ; S n − ). Then, thanks to Step 3, for H -a.e. r ∈ Iω n − d (cid:48) E ( r ) = ddr (cid:20) b E ( r )(sin α v ( r )) n − (cid:21) = b (cid:48) E ( r )(sin α v ( r )) n − + b E ( r ) ddr (cid:20) α v ( r )) n − (cid:21) = ( n − α (cid:48) v ( r ) cos α v ( r )(sin α v ( r )) n b E ( r ) + b E ( r ) (cid:20) − n − α v ( r )) n (cos α v ( r )) α (cid:48) v ( r ) (cid:21) = 0 , for H -a.e. r ∈ I . This shows that d E is H -a.e. constant in I . Therefore, E ∩ Φ( I × S n − )can be obtained by applying an orthogonal transformation to F v ∩ Φ( I × S n − ). (cid:3) Proof of Theorem 1.2: (i) = ⇒ (ii) We start by showing that the fact that { < α ∧ ≤ α ∨ < π } is an interval is a necessarycondition for rigidity. Proposition 8.1.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) ,such that F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3) .Suppose that the set { < α ∧ ≤ α ∨ < π } is not an interval. That is, suppose that thereexists r ∈ { α ∧ = 0 } ∪ { α ∨ = π } such that (0 , r ) ∩ { < α ∧ ≤ α ∨ < π } (cid:54) = ∅ and ( r, ∞ ) ∩ { < α ∧ ≤ α ∨ < π } (cid:54) = ∅ . Then, rigidity fails. More precisely, setting E := F v ∩ B ( r ) and E := F v \ B ( r ) , we have E ∪ ( RE ) ∈ N ( v ) for every R ∈ O ( n ) . Before giving the proof of Proposition 8.1 we need the following lemma. emma 8.2. Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) , such that F v is a set of finite perimeter and finite volume. Let α v be defined by (1.3) , and let r > .Then, ( ∂ ∗ F v ) r = H n − B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( re ) . Proof.
We divide the proof in two steps.
Step 1:
We show that ( ∂ ∗ F v ) r ⊂ B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( re ) . To this aim, it will be enough to show that α ∧ v ( r ) ≤ dist S n − (ˆ x, e ) ≤ α ∨ v ( r ) for every x ∈ ( ∂ ∗ F v ) r . (8.1)Let us first prove thatdist S n − (ˆ x, e ) ≤ α ∨ v ( r ) for every x ∈ ( ∂ ∗ F v ) r (8.2)Note that (8.2) is trivial if α ∨ v ( r ) = π . For this reason, we will assume α ∨ v ( r ) < π . Notenow that (8.2) follows if we prove that x ∈ ∂B ( r ) and dist S n − (ˆ x, e ) > α ∨ v ( r ) = ⇒ x ∈ F (0) v . (8.3)Let now x ∈ ∂B ( r ), and suppose that there exists δ > S n − (ˆ x, e ) = α ∨ v ( r ) + δ. Let now ρ > S n − (ˆ y, ˆ x ) < δ y ∈ B ( x, ρ ) . By triangle inequality for the geodesic distance we have, in particular, that α ∨ v ( r ) + δ = dist S n − (ˆ x, e ) ≤ dist S n − (ˆ x, ˆ y ) + dist S n − (ˆ y, e ) < δ S n − (ˆ y, e ) , so that dist S n − (ˆ y, e ) > α ∨ v ( r ) + δ y ∈ B ( x, ρ ) . (8.4)Thanks to the inequality above, by definition of F v we have F v ∩ B ( x, ρ ) ⊂ (cid:26) y ∈ R n : α ∨ v ( r ) + δ < dist S n − (ˆ y, e ) < α v ( | y | ) (cid:27) ∩ B ( x, ρ ) . Therefore, for every ρ ∈ (0 , ρ ) H n ( F v ∩ B ( x, ρ )) = ˆ r + ρr − ρ H n − ( F v ∩ B ( x, ρ ) ∩ ∂B ( r )) dr ≤ ˆ r + ρr − ρ χ { α v >α ∨ v ( r )+ δ/ } ( r ) H n − ( F v ∩ B ( x, ρ ) ∩ ∂B ( r )) dr = ˆ ( r − ρ,r + ρ ) ∩{ α v >α ∨ v ( r )+ δ/ } H n − ( F v ∩ B ( x, ρ ) ∩ ∂B ( r )) dr. Note now that, for ρ small enough, there exists C = C ( r ) > B ( x, ρ ) ∩ ∂B ( r ) ⊂ B Cρ ( r ˆ x ) for every r ∈ ( r − ρ, r + ρ ) . herefore, H n ( F v ∩ B ( x, ρ )) ≤ ˆ ( r − ρ,r + ρ ) ∩{ α v >α ∨ v ( r )+ δ/ } H n − ( B Cρ ( r ˆ x )) dr = ( n − ω n − ˆ ( r − ρ,r + ρ ) ∩{ α v >α ∨ v ( r )+ δ/ } r n − ˆ Cρ (sin τ ) n − dτ dr ≤ ( n − ω n − ˆ ( r − ρ,r + ρ ) ∩{ α v >α ∨ v ( r )+ δ/ } r n − ˆ Cρ τ n − dτ dr = ω n − C n − ( r + ρ ) n − ρ n − H (( r − ρ, r + ρ ) ∩ { α v > α ∨ v ( r ) + δ/ } ) . Thus, recalling the definition of α ∨ v ( r ),lim ρ → + H n ( F v ∩ B ( x, ρ )) ω n ρ n ≤ ω n − C n − ω n ( r + ρ ) n − lim ρ → + H (( r − ρ, r + ρ ) ∩ { α v > α ∨ v ( r ) + δ/ } ) ρ = 0 , which gives (8.3) and, in turn, (8.2). By similar arguments, one can prove that x ∈ ∂B ( r ) and dist S n − (ˆ x, e ) < α ∧ v ( r ) = ⇒ x ∈ F (1) v , which implies that α ∧ v ( r ) ≤ dist S n − (ˆ x, e ) for every x ∈ ( ∂ ∗ F v ) r . The above inequality, together with (8.2), shows (8.1).
Step 2:
We conclude. Thanks to Corollary 5.2, H n − (( ∂ ∗ F v ) r ) = H n − ( ∂ ∗ F v ∩ ∂B ( r )) = P ( F v ; ∂B ( r )) = r n − ( ξ ∨ v ( r ) − ξ ∧ v ( r ))= v ∨ ( r ) − v ∧ ( r ) = H n − ( B α ∨ v ( r ) ( re )) − H n − ( B α ∧ v ( r ) ( re ))= H n − (cid:16) B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( re ) (cid:17) Since, by Step 1, ( ∂ ∗ F v ) r ⊂ B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( re ) , we have( ∂ ∗ F v ) r = H n − B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( re ) = H n − B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( re ) . (cid:3) We can now give the proof of Proposition 8.1.
Proof of Proposition 8.1.
Note that, since B ( r ) is open and E ∩ B ( r ) = F v ∩ B ( r ), we have E ( t ) ∩ B ( r ) = ( E ∩ B ( r )) ( t ) = ( F v ∩ B ( r )) ( t ) = F ( t ) v ∩ B ( r ) for every t ∈ [0 , . From this, it follows that ∂ ∗ E ∩ B ( r ) = ∂ ∗ F v ∩ B ( r ) . (8.5)Similarly, we obtain ∂ ∗ E \ B ( r ) = ∂ ∗ ( RF v ) \ B ( r ) = ( R ∂ ∗ F v ) \ ( RB ( r )) = R ( ∂ ∗ F v \ B ( r )) . (8.6)Thus, thanks to (8.5) and (8.6) P ( E ) = H n − ( ∂ ∗ E ∩ B ( r )) + H n − ( ∂ ∗ E ∩ ∂B ( r )) + H n − ( ∂ ∗ E \ B ( r ))= H n − ( ∂ ∗ F v ∩ B ( r )) + H n − ( ∂ ∗ E ∩ ∂B ( r )) + H n − (cid:16) R ( ∂ ∗ F v \ B ( r )) (cid:17) = H n − ( ∂ ∗ F v ∩ B ( r )) + H n − ( ∂ ∗ E ∩ ∂B ( r )) + H n − ( ∂ ∗ F v \ B ( r )) . herefore, in order to conclude the proof we only need to show that H n − ( ∂ ∗ E ∩ B ( r )) = H n − ( ∂ ∗ F v ∩ B ( r )) . (8.7)Without any loss of generality, we will assume that α ∨ v ( r ) = ap lim( f, (0 , r ) , r ) , α ∧ v ( r ) = ap lim( f, ( r, ∞ ) , r ) . (8.8)Let now E , E , and R be as in the statement. We divide the proof of (8.7) into steps. Step 1:
We show that ( ∂ ∗ E ) r ⊂ B α ∨ v ( r ) ( re ) ∪ { R ( re ) } . To this aim, it will be enough to prove thatdist S n − (ˆ x, e ) ≤ α ∨ v ( r ) for every x ∈ ( ∂ ∗ E ) r . (8.9)If α ∨ v ( r ) = π inequality (8.9) is obvious, so we will assume that α ∨ v ( r ) < π . Step 1a:
We show that x ∈ ∂B ( r ) and dist S n − (ˆ x, e ) > α ∨ v ( r ) = ⇒ x ∈ E (0)1 . Indeed, let x ∈ ∂B ( r ), and suppose that there exists δ > S n − (ˆ x, e ) = α ∨ v ( r ) + δ. By repeating the argument used to show (8.4), we can choose ρ > S n − (ˆ y, e ) > α ∨ v ( r ) + δ y ∈ B ( x, ρ ) . By definition of E , we then have E ∩ B ( x, ρ ) = F v ∩ B ( r ) ∩ B ( x, ρ ) ⊂ (cid:26) y ∈ R n : | y | < r and α ∨ v ( r ) + δ < dist S n − (ˆ y, e ) < α v ( | y | ) (cid:27) ∩ B ( x, ρ ) . Therefore, for every ρ ∈ (0 , ρ ), by repeating the calculations done in Step 1 of Lemma 8.2,we obtain lim ρ → + ω n ρ n H n ( E ∩ B ( x, ρ ))= lim ρ → + ω n ρ n ˆ rr − ρ H n − ( F v ∩ B ( x, ρ ) ∩ ∂B ( r )) dr ≤ ω n − C n − ω n ( r + ρ ) n − lim ρ → + H (( r − ρ, r ) ∩ { α v > α ∨ v ( r ) + δ/ } ) ρ = 0 , where we used (8.8). Step 1b:
We show that ∂B ( r ) \ { R ( re ) } ⊂ ( RE ) (0) . Indeed, let x ∈ ∂B ( r ), and suppose that η := dist S n − (ˆ x, Re ) >
0. We are going to provethat x ∈ ( RE ) (0) . By repeating the argument used to show (8.4), we can choose ρ > S n − (ˆ y, Re ) > η y ∈ B ( x, ρ ) . Then, ( RE ) ∩ B ( x, ρ ) = (cid:16) R ( F v \ B ( r )) (cid:17) ∩ B ( x, ρ ) ⊂ H n (cid:110) y ∈ R n : | y | > r and η < dist S n − (ˆ y, Re ) < α v ( | y | ) (cid:111) ∩ B ( x, ρ ) . or ρ small enough, there exists C = C ( r ) > B ( x, ρ ) ∩ ∂B ( r ) ⊂ B Cρ ( r ˆ x ) for every r ∈ ( r − ρ, r + ρ ) . Therefore, for every ρ ∈ (0 , ρ ), H n (( RE ) ∩ B ( x, ρ )) ≤ ˆ ( r,r + ρ ) ∩{ α v >η/ } H n − ( B Cρ ( r ˆ x )) dr = ( n − ω n − ˆ ( r,r + ρ ) ∩{ α v >η/ } r n − ˆ Cρ (sin τ ) n − dτ dr = ω n − C n − ( r + ρ ) n − ρ n − H (( r, r + ρ ) ∩ { α v > η/ } ) . From this, thanks to (8.8), we obtainlim ρ → + H n (( RE ) ∩ B ( x, ρ )) ω n ρ n ≤ ω n − C n − ω n ( r + ρ ) n − lim ρ → + H (( r, r + ρ ) ∩ { α v > η/ } ) ρ = 0 . Step 1c:
We conclude the proof of Step 1. By definition of E , from Step 1a and Step 1bit follows that { x ∈ ∂B ( r ) : dist S n − (ˆ x, e ) > α ∨ v ( r ) } \ { Re } ⊂ E (0)1 ∩ ( RE ) (0) = E (0) . Therefore, ( ∂ ∗ E ) r ⊂ ∂B ( r ) \ (cid:0) { x ∈ ∂B ( r ) : dist S n − (ˆ x, e ) > α ∨ v ( r ) } \ { Re } (cid:1) = B α ∨ v ( r ) ( re ) ∪ { Re } . Step 2:
We show (8.7), concluding the proof. Thanks to Step 1 and Lemma 8.2 we have P ( E ; ∂B ( r )) = H n − ( ∂ ∗ E ∩ ∂B ( r )) = H n − (( ∂ ∗ E ) r ) ≤ H n − (cid:0) B α ∨ v ( r ) ( re ) (cid:1) = H n − ( ∂ ∗ F v ∩ ∂B ( r )) = P ( F v ; ∂B ( r )) ≤ P ( E ; ∂B ( r )) , where we also used (1.4) with B = { r } . (cid:3) We now show that, if the jump part D j α v of Dα v is non zero, rigidity fails. Proposition 8.3.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) suchthat F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3) .Suppose that α v has a jump at some point r > . Then, rigidity fails. More precisely,setting E := F v ∩ B ( r ) and E := F v \ B ( r ) , we have E ∪ ( RE ) ∈ N ( v ) , for every R ∈ O ( n ) such that < dist S n − ( Re , e ) < λ ( α ∨ v ( r ) − α ∧ v ( r )) for some λ ∈ (0 , . (8.10) Proof.
Let R ∈ O ( n ), λ ∈ (0 , E ∈ R n be as in the statement, and set ω := Re .Arguing as in the proof of Proposition 8.1 we have: P ( E ) = H n − ( ∂ ∗ F v ∩ B ( r )) + H n − ( ∂ ∗ E ∩ ∂B ( r )) + H n − ( ∂ ∗ F v \ B ( r )) . Therefore, in order to conclude the proof we only need to show that H n − ( ∂ ∗ E ∩ ∂B ( r )) = H n − ( ∂ ∗ F v ∩ ∂B ( r )) . (8.11)Without any loss of generality, we will assume that α ∨ v ( r ) = ap lim( f, (0 , r ) , r ) , α ∧ v ( r ) = ap lim( f, ( r, ∞ ) , r ) . (8.12)We now proceed by steps. tep 1: We show that ( ∂ ∗ E ) r ⊂ B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( rω ) . (8.13)To show (8.13), it is enough to prove that for every x ∈ ( ∂ ∗ E ) r we havedist S n − (ˆ x, e ) ≤ α ∨ v ( r ) for every x ∈ ( ∂ ∗ E ) r , (8.14)and dist S n − (ˆ x, ω ) ≥ α ∧ v ( r ) for every x ∈ ( ∂ ∗ E ) r . (8.15)We will only show (8.14), since (8.15) can be obtained in a similar way. Note that (8.14)is automatically satisfied if α ∨ v ( r ) = π , so we will assume α ∨ v ( r ) < π .By arguing as in Step 1a of the proof of Proposition 8.1 we obtain x ∈ ∂B ( r ) and dist S n − (ˆ x, e ) > α ∨ v ( r ) = ⇒ x ∈ E (0)1 . (8.16)Let us now prove that x ∈ ∂B ( r ) and dist S n − (ˆ x, e ) > α ∨ v ( r ) = ⇒ x ∈ ( R E ) (0) . (8.17)Let x ∈ ∂B ( r ), and suppose that there exists δ > S n − (ˆ x, e ) = α ∨ v ( r ) + δ. Thanks to the argument we used to show (8.4), we can choose ρ > S n − (ˆ y, e ) > α ∨ v ( r ) + δ y ∈ B ( x, ρ ) . Therefore, for every y ∈ B ( x, ρ ) we have α ∨ v ( r ) + δ < dist S n − (ˆ y, e ) ≤ dist S n − (ˆ y, ω ) + dist S n − ( ω, e ) < dist S n − (ˆ y, ω ) + λ ( α ∨ v ( r ) − α ∧ v ( r )) . Since r is a jump point for α v , we have α ∨ v ( r ) > α ∧ v ( r ), and the above inequality impliesthatdist S n − (ˆ y, ω ) > (1 − λ ) α ∨ v ( r ) + λα ∧ v ( r ) + δ > (1 − λ ) α ∧ v ( r ) + λα ∧ v ( r ) + δ α ∧ v ( r ) + δ , for every y ∈ B ( x, ρ ). Then, by definition of E ,( RE ) ∩ B ( x, ρ ) = (cid:16) R ( F v \ B ( r )) (cid:17) ∩ B ( x, ρ ) ⊂ H n (cid:26) y ∈ R n : | y | > r and α ∧ v ( r ) + δ < dist S n − (ˆ y, ω ) < α v ( | y | ) (cid:27) ∩ B ( x, ρ ) . As already observed in the previous proofs, for ρ small enough there exists C = C ( r ) > B ( x, ρ ) ∩ ∂B ( r ) ⊂ B Cρ ( r ˆ x ) for every r ∈ ( r − ρ, r + ρ ) . Therefore, for every ρ ∈ (0 , ρ ) sufficiently small H n (( RE ) ∩ B ( x, ρ )) ≤ ˆ ( r,r + ρ ) ∩{ α v >α ∧ v ( r )+ δ/ } H n − ( B Cρ ( r ˆ x )) dr = ( n − ω n − ˆ ( r,r + ρ ) ∩{ α v >α ∧ v ( r )+ δ/ } r n − ˆ Cρ (sin τ ) n − dτ dr = ω n − C n − ( r + ρ ) n − ρ n − H (( r, r + ρ ) ∩ { α v > α ∧ v ( r ) + δ/ } ) . rom this, thanks to (8.12), we obtainlim ρ → + H n (( RE ) ∩ B ( x, ρ )) ω n ρ n ≤ ω n − C n − ω n ( r + ρ ) n − lim ρ → + H (( r, r + ρ ) ∩ { α v > α ∧ v ( r ) + δ/ } ) ρ = 0 , which shows (8.17). This, together with (8.16), implies (8.14). As already mentioned,(8.15) can be proved in a similar way, and therefore (8.13) follows. Step 2:
We conclude. From (8.10) it follows that B α ∧ v ( r ) ( rω ) ⊂ B α ∨ v ( r ) ( re ) . Therefore, thanks to (8.13) and Lemma 8.2 P ( E ; ∂B ( r )) = H n − ( ∂ ∗ E ∩ ∂B ( r )) = H n − (( ∂ ∗ E ) r ) ≤ H n − (cid:0) B α ∨ v ( r ) ( re ) \ B α ∧ v ( r ) ( rω ) (cid:1) = v ∨ ( r ) − v ∧ ( r ) = P ( F v ; ∂B ( r )) ≤ P ( E ; ∂B ( r )) , where we also used (1.4) with B = { r } . Then, (8.11) follows from the last chain ofinequalities. (cid:3) We conclude this section showing that, if D c α v (cid:54) = 0, rigidity fails. Proposition 8.4.
Let v : (0 , ∞ ) → [0 , ∞ ) be a measurable function satisfying (1.2) suchthat F v is a set of finite perimeter and finite volume, and let α v be defined by (1.3) . Supposethat D c α v (cid:54) = 0 . Then, rigidity fails.Proof. We are going to construct a spherically v -distributed set E ∈ N ( v ) that cannot beobtained by applying a single orthogonal transformation to F v (see (8.20) below).First of all, let us note that it is not restrictive to assume that α v is purely Cantorian.Indeed, by (2.4) one can decompose α v into α v = α av + α jv + α cv , (8.18)where α av ∈ W , (0 , ∞ ), α jv is a purely jump function, and α cv is purely Cantorian. Thanksto (8.18), in the general case when α v (cid:54) = α cv , the proof can be repeated by applying ourargument just to the Cantorian part α cv of α v . Therefore, from now on we will assumethat Dα v = D c α v . Thanks to Proposition 8.1, we can also assume that { < α ∧ v ≤ α ∨ v < π } is an interval(otherwise there is nothing to prove, since rigidity fails). Moreover, since α v is continuous,there exist a, b >
0, with a < b , such that I := ( a, b ) ⊂⊂ { < α ∧ v ≤ α ∨ v < π } and0 < α v ( r ) < π for every r ∈ I. (8.19)Since D c α v (cid:54) = 0, it is not restrictive to assume | D c α v | ( I ) >
0. For each γ ∈ ( − π, π ), wedefine R γ ∈ O ( n ) in the following way: R γ x x x ... x n = x cos γ − x sin γx sin γ + x cos γx ... x n . hat is, R γ is a counterclockwise rotation of the angle γ in the plane ( x , x ). Let now fix λ ∈ (0 , β : (0 , ∞ ) → ( − π, π ) as β ( r ) := r ∈ (0 , a ) ,λ ( α v ( r ) − α v ( a )) if r ∈ [ a, b ] ,λ ( α v ( b ) − α v ( a )) if r ∈ ( b, ∞ ) . We set E := { x ∈ R n : dist S n − (ˆ x, R β ( | x | ) e ) < α ∨ v ( | x | ) } . (8.20)Clearly, E cannot be obtained by applying a single orthogonal transformation to F v . Letus show that E ∈ N ( v ), so that rigidity fails. We proceed by steps. Step 1:
We construct a sequence of functions v k : I → [0 , ∞ ) satisfying the followingproperties:(a) lim k →∞ α v k ( r ) = α v ( r ) for H -a.e. r ∈ I ;(b) Dξ v k = D j ξ v k for every k ∈ N ;(c) lim k →∞ P ( F v k ; Φ( I × S n − )) = P ( F v ; Φ( I × S n − )).First of all note that, by (3.5) and by the chain rule in BV (see, [2, Theorem 3.96]), itfollows that ξ v is purely Cantorian, where ξ v is given by (3.3). Moreover, from (2.5) andfrom the fact that ξ v is continuous, we have | Dξ v | ( I ) = sup (cid:40) N − (cid:88) i =1 | ξ v ( r i +1 ) − ξ v ( r i ) | : a < r < r < . . . < r N < b (cid:41) , where the supremum runs over N ∈ N and over all r , . . . , r N with a < r < r <. . . < r N < b . Therefore, for every k ∈ N there exist N k ∈ N and r k , . . . , r kN with a < r k < r k < . . . < r kN < b such that | Dξ v | ( I ) ≤ N k − (cid:88) i =1 | ξ v ( r ki +1 ) − ξ v ( r ki ) | + 1 k and | r ki +1 − r ki | < k for every i = 1 , . . . , N k − . Without any loss of generality, we can assume that the partitions are increasing in k . Thatis, we will assume that { r k , . . . , r kN k } ⊂ { r k +11 , . . . , r k +1 N k +1 } for every k ∈ N . Define now, for every k ∈ N , ξ kv ( r ) := N k (cid:88) i =0 ξ v ( r ki ) χ [ r ki ,r ki +1 ) ( r ) , (8.21)where we set r k := a and r kN k +1 := b . Let us now set v k ( r ) := ξ kv ( r ) /r n − for every r ∈ I and for every k ∈ N , and note that, by definition, ξ kv = ξ v k . Since ξ v is continuous, we have thatlim k →∞ ξ kv ( r ) = ξ v ( r ) for H -a.e. r ∈ I. (8.22)Recalling (3.5) and (3.6), last relation implies property (a). Moreover, from (8.21) we have(b). et us now show (c). Thanks to (8.19) and (8.22), we havelim k →∞ p F kv ( r ) = p F v ( r ) for H -a.e. r ∈ I. (8.23)Moreover, | Dξ kv | ( I ) = N k (cid:88) i =0 | ξ v ( r ki +1 ) − ξ v ( r ki ) | (8.24)= | ξ v ( r k ) − ξ v ( a ) | + | ξ v ( b ) − ξ v ( r kN k ) | + N k − (cid:88) i =1 | ξ v ( r ki +1 ) − ξ v ( r ki ) | . Since | Dξ v | ( I ) − k ≤ N k − (cid:88) i =1 | ξ v ( r ki +1 ) − ξ v ( r ki ) | ≤ | Dξ v | ( I ) , using (8.24) and the fact that ξ v is continuous we obtain | Dξ v | ( I ) = lim k →∞ N k − (cid:88) i =1 | ξ v ( r ki +1 ) − ξ v ( r ki ) | = lim k →∞ | Dξ kv | ( I ) . (8.25)Thanks to [2, Theorem 3.23], up to subsequences ξ kv weakly* converges in BV ( I ) to ξ v .Since, in addition, (8.25) holds true, we can apply [2, Proposition 1.80] to the sequence ofmeasures {| Dξ kv |} k ∈ N . Therefore, recalling that Dξ kv = D s ξ kv and Dξ v = D s ξ v , we havelim k →∞ ˆ I r n d | D s ξ kv | ( r ) = lim k →∞ ˆ I r n d | Dξ kv | ( r ) = ˆ I r n d | Dξ v | ( r ) = ˆ I r n d | D s ξ v | ( r ) . Then, from Corollary 5.2lim k →∞ P ( F v k ; Φ( I × S n − )) = lim k →∞ (cid:18) ˆ I p F vk ( r ) dr + ˆ I r n − d | D s ξ kv | ( r ) (cid:19) = (cid:18) ˆ I p F v ( r ) dr + ˆ I r n − d | D s ξ v | ( r ) (cid:19) = P ( F v ; Φ( I × S n − )) , where we also used (8.23). Step 2:
For each k ∈ N , we construct a spherically v k -distributed set E k such that P ( E k ; Φ( I × S n − )) = P ( F v k ; Φ( I × S n − )) . From (3.5) and (3.6) it follows that α v k = F − ( ξ kv ) ∈ BV ( I ), and α v k ( r ) = N k (cid:88) i =0 α v ( r ki ) χ [ r ki ,r ki +1 ) ( r ) . (8.26)Therefore, for each k ∈ N we have that Dα v k = D j α v k , and the jump set of α v k is a finiteset. More precisely, Dα v k = N k (cid:88) i =1 ( α v ( r ki ) − α v ( r ki − )) δ r ki , where δ r denotes the Dirac delta measure concentrated at r . Let λ ∈ (0 ,
1) be fixed, anddefine the set E k ⊂ Φ( I × S n − ) as E k := (cid:104) F v k ∩ ( B ( r k ) \ B ( a )) (cid:105) ∪ (cid:104) R λ ( α v ( r k ) − α v ( a )) ( F v k ∩ ( B ( b ) \ B ( r k ))) (cid:105) . Thanks to Proposition 8.3, we have that P ( E k ; Φ( I × S n − )) = P ( F v k ; Φ( I × S n − )) . efine now E k ⊂ Φ( I × S n − ) as E k := ( E k ∩ B ( r k )) ∪ (cid:104) R λ ( α v ( r k ) − α v ( r k )) ( E k \ B ( r k )) (cid:105) . Applying again Proposition 8.3, we have P ( E k ; Φ( I × S n − )) = P ( E k ; Φ( I × S n − )) = P ( F v k ; Φ( I × S n − )) . Note that, since R γ is associative with respect to γ (that is, we have R γ R γ = R γ + γ ),we can write E k as E k = (cid:104) F v k ∩ ( B ( r k ) \ B ( a )) (cid:105) ∪ (cid:104) R λ ( α v ( r k ) − α v ( a )) ( F v k ∩ ( B ( r k ) \ B ( r k ))) (cid:105) ∪ (cid:104) R λ ( α v ( r k ) − α v ( a )) ( F v k ∩ ( B ( b ) \ B ( r k ))) (cid:105) . Iterating this procedure N k times, we obtain that P ( E k ; Φ( I × S n − )) = P ( F v k ; Φ( I × S n − )) , where E k := E kN k = { x ∈ Φ( I × S n − ) : dist S n − (ˆ x, R λ ( α vk ( | x | ) − α vk ( a ) ) e ) < α v k ( | x | ) } . (8.27) Step 3:
We show that E k −→ (cid:98) E in Φ( I × S n − ), for some spherically v -distributed set (cid:98) E such that P ( (cid:98) E ; Φ( I × S n − )) = P ( F v ; Φ( I × S n − )) . From (8.26) and (8.22) it follows thatlim k →∞ α v k ( r ) = α v ( r ) for H -a.e. r ∈ I. Therefore, from (8.27) we have E k −→ (cid:98) E ( in (Φ( I × S n − ))), where (cid:98) E is the spherically v -distributed set in Φ( I × S n − ) given by (cid:98) E := { x ∈ Φ( I × S n − ) : dist S n − (ˆ x, R λ ( α v ( | x | ) − α v ( a )) e ) < α v ( | x | ) } . (8.28)Then, by the lower semicontinuity of the perimeter with respect to the L convergence(see, for instance, [22, Proposition 12.15]): P ( (cid:98) E ; Φ( I × S n − )) ≤ lim k →∞ P ( E k ; Φ( I × S n − ))lim k →∞ P ( F v k ; Φ( I × S n − )) = P ( F v ; Φ( I × S n − )) ≤ P ( (cid:98) E ; Φ( I × S n − )) , where we also used (1.4). Step 4:
We conclude. Let E be given by (8.20). Then, E is spherically v -distributed andsatisfies E = H n ( F v ∩ ( B ( a ))) ∪ (cid:104) (cid:98) E ∩ ( B ( b ) \ B ( a )) (cid:105) ∪ (cid:2) R λ ( α v ( b ) − α v ( a )) ( F v \ ( B ( b ))) (cid:3) , here (cid:98) E is defined in (8.28). By repeating the arguments used in the proof of Proposi-tion 8.1, and using the fact that Φ( I × S n − ) = B ( b ) \ B ( a ), one can see that P ( E ) = P ( E ; B ( a )) + P ( E ; ∂B ( a )) + P ( E ; B ( b ) \ B ( a ))+ P ( E ; ∂B ( b )) + P ( E ; R n \ B ( b ))= P ( F v ; B ( a )) + P ( E ; ∂B ( a )) + P ( (cid:98) E ; B ( b ) \ B ( a ))+ P ( E ; ∂B ( b )) + P ( F v ; R n \ B ( b ))= P ( F v ; B ( a )) + P ( E ; ∂B ( a )) + P ( F v ; B ( b ) \ B ( a ))+ P ( E ; ∂B ( b )) + P ( F v ; R n \ B ( b )) , where we also used Step 3 and the invariance of the perimeter under orthogonal trans-formations. Since α v is continuous, an argument similar to the one used to prove (8.13)shows that P ( E ; ∂B ( a )) = P ( E ; ∂B ( b )) = 0 . Therefore, P ( E ) = P ( F v ; B ( a )) + P ( F v ; B ( b ) \ B ( a )) + P ( F v ; R n \ B ( b )) = P ( F v ) . (cid:3) We can now give the proof of the implication (i) = ⇒ (ii) of Theorem 1.2. Proof of Theorem 1.2: (i) = ⇒ (ii). To show the implication, it suffices to combine Propo-sition 8.1, Proposition 8.3, and Proposition 8.4. (cid:3) Acknowledgements
The authors would like to thank Marco Cicalese, Nicola Fusco, and Emanuele Spadarofor inspiring discussions on the subject. They would also like to thank Frank Morganfor useful comments on a preliminary version of the paper. F. Cagnetti was supportedby the EPSRC under the Grant EP/P007287/1 “Symmetry of Minimisers in Calculus ofVariations”.
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E-mail address : [email protected] Department of Mathematics, University of Sussex, Pevensey 2, BN1 9QH, Brighton, UK
E-mail address : [email protected] Technische Universit¨at M¨unchen, Zentrum Mathematik - M15, Boltzmannstrasse 3, 85747Garching, Germany
E-mail address : [email protected]@ma.tum.de