THE STABILITY OF THE ISOPERIMETRIC INEQUALITY ... - CMU Math

THE STABILITY OF THE ISOPERIMETRIC INEQUALITY N.FUSCO

These notes contain five lectures given by the author at the CNA Summer School held at Carnegie Mellon University in Pittsburgh from May 30 to June 7, 2013. The aim of the course was to give a self contained introduction to the classical isoperimetric inequality and to various stability results proved in recent years for this inequality and other related geometric and analytic inequalities. In the first lecture we recall the basic definitions and the main properties of sets of finite perimeter: the structure theorem, approximation with smooth sets, compactness. These are the main ingredients of De Giorgi’s proof of the isoperimetric inequality via Steiner symmetrization. This result is established in the second lecture which also contains the coarea formula for both functions and sets of finite perimeter and the proof of the equivalence between isoperimetric and Sobolev inequality. With the third lecture we enter into the subject of the stability. There, we give the complete proof of the quantitative estimates obtained by Fuglede for convex and nearly spherical sets. In Lecture 4 we give a detailed account of the proof of the quantitative isoperimetric inequality for general sets first obtained by F., Maggi and Pratelli. This proof is based on symmetrization arguments that also apply to a variety of other inequalities. In the last lecture two different proofs are presented. The first one, due to Figalli, Maggi and Pratelli starts from Gromov’s proof of the isoperimetric property of the balls and uses Brenier optimal transportation map between sets. The second one, due to Cicalese and Leonardi, is based on the regularity theory for area almost minimizers. Most of the notes were taken by Ryan Murray who typed them live in the classroom. Matteo Rinaldi added some extra material from some hand written notes of mine and did the editing. Laura Bufford did all the pictures. I warmly thank them for the excellent job. Without their precious help probably these notes would have never appeared. 1. First lecture: sets of finite perimeter and reduced boundary The isoperimetric inequality is one of the most celebrated results of modern mathematics with countless applications to all areas of mathematics. Since the first two-dimensional proof of Hurwitz ([37] [38]) appeared at the beginning of the last century, many different proofs of this inequality have been given under different regularity assumptions on the set E. In its utmost generality it was proven by De Giorgi in [18] in the framework of the theory of sets of finite perimeter. It states that if E is a measurable set in Rn with |E| = |BR |, then P (E) ≥ P (BR ), with the equality holding if and only if E is a ball. Here, P (E) denotes the perimeter of the set E. In order to prove this important inequality we start by recalling the definitions and the main properties of sets of finite perimeter. As a reference for the results stated here one may look at [20, Chap. 5], [2, Chap. 3] or at the original papers of De Giorgi collected in [19]. 1

Definition 1.1. Let Ω be an open set in Rn . The perimeter of E in Ω is defined as Z  1 n P (E; Ω) = sup divϕ dx : ϕ ∈ Cc (Ω, R ), ||ϕ||∞ ≤ 1 . E

Note that this definition is unaffected by modifications on sets of measure zero. Thus the two sets shown in the picture below have the same perimeter.

Figure 1. Two sets with the same perimeter If E is smooth the classical divergence implies that Z Z divϕ dx = E

ϕ · νdHn−1 ,

∂E∩Ω

where Hk stands for the k-dimensional Hausdorff measure in Rn . Thus, taking the supremum over all functions ϕ ∈ Cc1 (Ω, Rn ), ||ϕ||∞ ≤ 1, we get P (E; Ω) = Hn−1 (∂E ∩ Ω). Furthermore, if P (E; Ω) < ∞ the map ϕ∈

Cc1 (Ω, Rn )

Z 7→

divϕ dx E

is linear and continuous with respect to the uniform convergence on Cc1 (Ω, Rn ). Therefore Riesz’s theorem yields that there exists a vector valued Radon measure µ = (µ1 , . . . , µn ) in Ω such that Z Z Z n Z X χE divϕ dx = divϕ dx = ϕ · dµ = ϕi dµi Ω

E

Ω

i=1

Ω

for all ϕ ∈ Cc1 (Ω, Rn ). Thus µ = −DχE , where DχE is the distributional derivative of χE and the above formula can be rewritten as Z Z (1) divϕ dx = − ϕ · dDχE . E

Ω

In conclusion, E has finite perimeter in Ω if and only if DχE is a Radon measure with values in Rn and finite total variation. In fact, from the Definition 1.1 we immediately get that P (E; Ω) = |DχE |(Ω). Rn

If Ω = we simply write P (E) in place of P (E; Rn ) and if P (E) < ∞ we say that E is a set of finite perimeter. 2

Exercise 1.2. Show that if E has finite perimeter in Ω, then ˚ ∩ Ω) = P (E; Ω \ E) ¯ = 0. P (E; E Note that from Exercise 1.2 it follows that DχE concentrates on ∂E ∩ Ω and (1) can be rewritten as Z Z (2) divϕ dx = − ϕ · DχE , ∀ ϕ ∈ Cc1 (Ω, Rn ). E

∂E∩Ω

From Besicovitch derivation theorem ([2, Th. 2.22]) we have that for |DχE |–a.e. x ∈ supp|DχE | there exists the derivative of DχE with respect to its total variation |DχE | and that it is a vector of length 1. For such points we have DχE (Br (x)) DχE (x) = lim =: −ν E (x) and |ν E (x)| = 1. (3) r→0 |DχE |(Br (x)) |DχE | Definition 1.3. We shall denote by ∂ ∗ E the set of all points in supp|DχE | where (3) holds. The set ∂ ∗ E is called the reduced boundary of E, while the vector ν E (x) is the generalized exterior normal at x. From (3) we have that the measure DχE can be represented by integrating −ν E with respect to |DχE |, i.e. DχE = −ν E |DχE |. Thus (2) can be rewritten as Z Z (4) divϕdx = ϕ · ν E d|DχE |, ∀ ϕ ∈ Cc1 (Ω, Rn ). ∂ ∗ E∩Ω

E

∂∗E

Since ⊂ supp|DχE | ⊂ ∂E, the reduced boundary of E is a subset of the topological boundary. Moreover, as a consequence of De Giorgi structure Theorem 1.7, if E has finite perimeter, then Hn−1 (∂ ∗ E) = P (E) < ∞. Next example shows that in general ∂ ∗ E can be much smaller than ∂E. Example 1.4. Let us take a sequence {qi } dense in Rn and set E :=

∞ [

B2−i (qi ). Observe that |∂E| = ∞.

i=1

Nevertheless E is a set of finite perimeter. To see this take ϕ ∈ Cc1 (Ω, Rn ), ||ϕ||∞ ≤ 1, and note that Z Z Z divϕ dx = lim S divϕ dx = lim ϕ · ν dHn−1 ≤ S E

N →∞

N i=1

B2−i (qi )

N →∞ ∂(

N i=1

B2−i (qi ))

N N ∞  [  X X
n−1 n−1 ≤ lim H ∂ B2−i (qi ) ≤ lim H ∂B2−i (qi ) = nωn 2−i(n−1) < +∞, N →∞

N →∞

i=1

i=1

i=1

where ωn denotes the measure of the unit ball. In dimension 1, sets of finite perimeter are easily characterized (see [2, Prop. 3.52]). Theorem 1.5. Let E ⊂ R be a measurable set. Then E has finite perimeter in R if and only if there exist −∞ ≤ a1 < b1 < a2 < b2 < . . . < bn ≤ +∞ such that n [ E= (ai , bi ) i=1

up to a Lebesgue measure zero set. Moreover, if Ω ⊂ R is an open set, P (E; Ω) = #({ai , bi ∈ Ω}). 3

Remark 1.6. We have to pay attention to the following fact: if E = (0, 1) ∪ (1, 2), then P (E) = 2 and ∂ ∗ E = {0, 2}. This is not surprising since, as we already observed, the definition of perimeter does not change if we modify E by a Lebesgue measure zero set. 1.1. Structure theorem for sets of finite perimeter. Next theorem deals with the structure of sets of sets of perimeter. For a proof see [20, Sec. 5.7.3] or [2, Th. 3.59]. Theorem 1.7 (De Giorgi). Let E ⊂ Rn be a measurable set of finite perimeter. Then the following hold: S (i) ∂ ∗ E is n − 1-rectifiable, i.e., ∂ ∗ E = i Ki ∪ N0 , where Hn−1 (N0 ) = 0 and Ki are compact subsets of C 1 manifolds Mi of dimension n − 1. (ii) |DχE | = Hn−1

∂ ∗ E;

(iii) for Hn−1 -a.e. x ∈ Ki , the generalized exterior normal ν E (x) is orthogonal to the tangent plane TxMi to the manifold Mi at x; |E ∩ Br (x)| 1 → , as r → 0; Br (x) 2 n−1 ∗ H (∂ E ∩ Br (x)) = 1. (v) ∀x ∈ ∂ ∗ E, lim r→0 ωn−1 rn−1

(iv) ∀x ∈ ∂ ∗ E,

Remark 1.8. From (ii) above we have that (4) can we rewritten as Z Z divϕ dx = ϕ · ν E dHn−1 , ∀ ϕ ∈ Cc1 (Rn , Rn ). E

∂E

Note that (iv) states that each point in ∂ ∗ E is a point of density

1 2

for E (see Definition 1.10 below).

Example 1.9. Let Q be a square in R2 . The reduced boundary is given by ∂ ∗ Q = ∂Q \

4 [

{vi }, where

i=1

vi are the vertices of the Q. As a matter of fact, for any sufficiently small ball B centered at any of the vi we have that |Q ∩ B|/|B| = 14 6= 21 .

Figure 2. Density of vertices Furthermore, property (v) tells us that if x ∈ ∂ ∗ E then the reduced boundary ∂ ∗ E looks flatter and flatter at small scales. Observe in fact that if we rescale ∂ ∗ E around x, we have 4

H

n−1



∂∗E − x ∩ B1 (0) r

 =

Hn−1 (∂ ∗ E ∩ Br (x)) → ωn−1 . rn−1

Figure 3. Rescaling around x Definition 1.10. Given E a measurable set and x ∈ Rn , the density of E at x, D(x; E) is defined as D(x; E) := lim

r→0

|E ∩ Br (x)| . ωn r n

If 0 ≤ a ≤ 1 we denote by E (a) the set of all points where the density of E is equal to a. Using this notation, part (iv) of De Giorgi’s Theorem 1.7 can be written as ∂ ∗ E ⊂ E (1/2) . Next result gives an even more precise description of what is going on with sets of finite perimeter. For the proof see [2, Th. 3.61]. Theorem 1.11 (Federer). If E ⊂ Rn is a set of finite perimeter, then - ∂ ∗ E ∈ E (1/2) and Hn−1 (E (1/2) \ ∂ ∗ E) = 0, - Hn−1 (Rn \ (E (0) ∪ E (1) ∪ E (1/2) ) = 0. Note that if E is a set of finite perimeter in Ω Theorems 1.7 and 1.11 hold in local form. Exercise 1.12. Let U ⊂ Rn−1 be a bounded open set and Ω = U × R. Let f : U → R be a Lipschitz function. Set Sf = {(x, t) ∈ Ω : t < f (x)}. Then show that - Sf has finite perimeter in Ω, - ∂ ∗ Sf coincides with Γf = {(x, f (x)) : x ∈ U} up to a set of Hn−1 measure zero set, (−∇f, 1) - ν Sf (x) coincides Hn−1 -a.e. on Γf with the standard normal p . 1 + |∇f |2 Exercise 1.13. Let f : R → R be the function f (x) = x2 sin x1 . Let E = {(x, y) ∈ R2 : y < f (x)} be the subgraph of f . Using the fact that f 0 (0) = 0, show that |E ∩ Br | 1 → . |Br | 2 Prove that (0, 0) ∈ / ∂ ∗ E by showing that lim sup r→0

Hn−1 (∂ ∗ E ∩ Br ) > 1. ωn−1 rn−1 5

Note that with Br we denote the ball with center at the origin. We conclude this section with few words on the approximation of sets of finite perimeter. Definition 1.14. Given a sequence of measurable sets Ej and a measurable set E, we say that Ej → E in measure in Ω if χEj → χE in L1 (Ω), i.e. |(Ej ∆E) ∩ Ω| → 0, as j → ∞. An important property of the perimeters is the lower semicontinuity with respect to the convergence in measure. This is a straightforward consequence of Definition 1.1. If Ej is a sequence of measurable sets converging in measure in Ω to E, then P (E; Ω) ≤ lim inf P (Ej ; Ω). j→∞

Next result, is the counterpart of Rellich-Kondrakov compactness theorem in the framework of sets of finite perimeter (see [2, Th. 3.39]). Theorem 1.15. Given a bounded open set Ω ⊂ Rn and a sequence of measurable sets Ej such that supj P (Ej ; Ω) < ∞, there exists a set E of finite perimeter in Ω such that, up to a subsequence, Ej → E in measure in Ω. Furthermore, we have the following approximation result ([2, Th. 3.42]). Theorem 1.16. Let E be a set of finite perimeter. Then there exists a sequence of smooth, bounded, open sets Ej such that Ej → E in measure in Rn and P (Ej ) → P (E). In view of this theorem and of the consequences of Definition 1.14 we have that E is a set of finite perimeter in Rn if and only if there exist a sequence of smooth open sets Ej ⊂ Rn , such that Ej → E in measure in Rn

and

sup P (Ej ) < ∞. j

Note also that in Theorem 1.16 one may replace the smooth sets Ej with polyhedra, i.e. bounded open sets obtained as the intersection of finitely many half–spaces. A local version of Theorem 1.16 is also true (see [2, Rem. 3.43]). Exercise 1.17. Using Theorem 1.16, show that if E and F are sets of finite perimeter, the same is true for E ∪ F , E ∩ F and E \ F and that P (E ∩ F ) + P (E ∪ F ) ≤ P (E) + P (F ).

2. Second lecture: isoperimetric and Sobolev inequalities We shall present De Giorgi’s proof of the isoperimetric inequality. Theorem 2.1. Let E ⊂ Rn be a measurable set with |E| = |BR |. Then (5)

P (BR ) ≤ P (E)

with the equality holding if and only if E is a ball. At the time De Giorgi proved this result above, the isoperimetric inequality was well known for sets E ⊂ Rn satisfying some additional regularity properties and several proofs were found, starting from the beginning of 20-th century. However, what is really striking is that the above result is proved in the utmost generality for any measurable set. De Giorgi’s proof follows an idea that Steiner had one century before ([46]). Actually, the proof of the isoperimetric property of the ball was the original motivation for Steiner to introduce what is now now known as Steiner symmetrization. 6

(a) Symmetrization wrt the plane {xn = 0}

(b) Symmetrization wrt a plane with normal ν 6= en

Figure 4. Steiner symmetral of a measurable set

Definition 2.2. Let E ⊂ Rn be a measurable set. For x ∈ Rn−1 set Ex := {y ∈ R : (x, y) ∈ E} and l(x) := H1 (Ex ). Then the Steiner symmetrization of E with respect to the hyperplane {xn = 0} is given by E s = {(x, y) ∈ Rn−1 × R : −l(x)/2 < y < l(x)/2}. The previous definition can be extended in an obvious way to any hyperplane πν passing through the origin and orthogonal to a unit vector ν. The resulting Steiner symmetrization of E with respect to πν will be denoted by E s,ν . The notation E s will be always used to denote the symmetrization of E with respect to {xn = 0}. By Fubini’s theorem we have immediately that |E| = |E s,ν |, while it is not too difficult to show ([20, Lemma 2, Sec. 2.2]) that diam(E s,ν ) ≤ diam(E), see Figure 4. Theorem 2.3. Let E be a set of finite perimeter with |E| < ∞, then the following hold: (i) (ii) (iii) (iv)

l ∈ BV (Rn−1 ), l ∈ W 1,1 (Rn−1 ) ⇔ Hn−1 ({z ∈ ∂ ∗ E : νnE (z) = 0}) = 0, P (E s ) ≤ P (E), P (E s ) = P (E) ⇒ for Hn−1 -a.e. x ∈ Rn−1 , Ex is a line segment up to a set of zero H1 measure.

Remark 2.4. Inequality (iii) is classical and is proved (for smooth sets) in the beautiful book of P´ olya– Szeg˝o [44], (iv) was proved in a weaker form by De Giorgi in his paper on the isoperimetric property of balls [18], (iv) in the form above, (i) and (ii) were proved by Chleb´ık–Cianchi–F. in [12, Th. 1.1, Lemma 3.1, Prop. 1.2]. Note that if P (E) = P (E s ), then E and E s are not necessarily equal up to a translation, as shown in Figure 5. In both pictures P (E) = P (E s ), conclusion (iv) of the theorem holds but E 6= E s up to a translation. 7

Figure 5. In general, symmetrals are not translated of the original sets

We now turn to the proof of the isoperimetric inequality via Steiner symmetrization. Proof. In order to prove (5) it is enough to show that given a set E such that |E| = |B1 | = ωn , then P (E) ≥ P (B1 ) with the equality holding if and only if E is a ball. Step 1. We first fix BR , with R >> 1 and consider the minimum problem min {P (E) : E ⊂ BR , |E| = ωn } . Let Ej ⊂ BR , with |Ej | = ωn , be a minimizing sequence for the above problem, i.e., limj P (Ej ) = inf{P (E) : E ⊂ BR , |E| = ωn }. By the compactness Theorem 1.15 we may assume, up to a not relabelled subsequence, that Ej converge in measure to some set F ⊂ BR , with |F | = ωn . By the lower semicontinuity of the perimeter we have P (F ) ≤ lim inf P (Ej ) and thus F is a minimizer. We claim that F is convex. To prove this, fix ν ∈ Sn−1 and consider the Steiner symmetrization F s,ν of F with respect to the hyperplane πν passing through the origin and orthogonal to ν (see Figure 4 (b) above). Since |F s,ν | = |F | = ωn , F s,ν ⊂ BR and P (F s,ν ) ≤ P (F ), by the minimality of F we have indeed P (F s,ν ) = P (F ). Therefore, by (iv) of Theorem 2.3, we have that for Hn−1 -a.e. x ∈ πν , the section {τ ∈ R : x + τ ν ∈ F } coincides (up to a set of H1 measure zero) with an open interval and this property holds for all directions ν. Notice that if we knew that for all ν and for all x ∈ πν , the section {τ ∈ R : x + τ ν ∈ F } is an open interval we would conclude at once that F is a convex set. However, using Lemma 2.5 below we may still conclude that F (1) , the set of all points where F has density 1, is a convex set, actually an open convex set. To simplify the notation let us set F = F (1) . Our goal now is to show that F is a ball. Denote by U the projection of F on Rn−1 . Then there exist two functions y1 , y2 : U → R, y1 convex and y2 concave, such that F = {(x, y) : x ∈ U, y1 (x) < y < y2 (x)} (see Figure 6 below). Moreover, F s = {(x, y) : x ∈ U, (y1 − y2 )(x)/2 < y < (y2 − y1 )(x)/2}. We have: Z p Z p 2 1 + |∇y1 | + 1 + |∇y2 |2 , P (F ) = U

U

Z p P (F ) = 2 1 + |∇(y2 − y1 )/2|2 . s

U 8

Figure 6. Projection of F √ Since F is a minimizer, P (F ) = P (F s ) and thus by the strict convexity of the function t 7→ 1 + t2 we get that ∇y2 = −∇y1 , hence y2 = −y1 + c, thus proving that F = F s up to a translation. Repeating this argument for all the Steiner symmetrizations F s,ν , with ν ∈ Sn−1 , we finally conclude that F must be a ball. This proves the isoperimetric inequality for a bounded set E. Step 2. Let us now consider an unbounded set E with |E| = ωn . From Theorem 1.16 we get a sequence of smooth bounded sets Ej such that Ej converge in measure to E in Rn and P (Ej ) → P (E) as j → ∞. From Step 1 we then have that P (Ej ) ≥ P (Brj ) where |Ej | = |Brj |. From this inequality and using the fact that |Ej | → |E|, letting j → ∞ we then have that P (E) ≥ P (B1 ). Finally, if P (E) = P (B1 ) we may repeat the same argument used in Step 1 to conclude first that E (1) is an open convex set and then that it is a ball.  Next result is proved in [28, Lemma 4.12]. Lemma 2.5. Let E be a measurable set in Rn such that for Hn−1 -a.e. x ∈ Rn−1 the section Ex = {y ∈ R : (x, y) ∈ E} is equivalent modulo H1 to a segment. Then, denoting by F the set of points of density 1 with respect to E, Fx is a segment for every x ∈ Rn−1 . An equivalent way of stating the isoperimetric inequality can be obtained observing that if |E| = |BR |, |E| = ωn Rn and P (BR ) = nωn Rn−1 and thus (5) becomes P (E) ≥ nωn1/n |E|1−1/n .

(6)

Note that this equality looks very similar to the Sobolev inequality since P (E) = |DχE |(Rn ) and n = |E|1−1/n . To understand better this connection we need to introduce an important kχE k n−1 n L

(R )

formula, first proved by Fleming and Rishel in [25]. As shown in the picture below, it is a sort of curvilinear version of the familiar Fubini theorem. Theorem 2.6 (Coarea formula for Lipschitz function). Let Ω ⊂ Rn be an open set and f : Ω → R a Lipschitz function. Then for a.e. t ∈ R, {f > t} is a set of finite perimeter. Moreover, if g : Ω → [0, +∞] is a Borel function, Z Z Z (7) g(x)|∇f |dx = dt g(x)dHn−1 (x). Ω

∂ ∗ {f >t}

R 9

Figure 7. Level sets Now we are ready to prove the following result. Theorem 2.7. The following statements are equivalent: n−1

(1) for all measurable set E with finite measure P (E) ≥ C0 |E| n ; n (2) for all f ∈ W 1,1 (Rn ) we have that k∇f kL1 (Rn ) ≥ C0 kf k n−1 . n L

(R )

Proof. To show that (2) ⇒ (1), i.e., that the Sobolev inequality implies the isoperimetric inequality, we use mollifiers. For ε > 0 set fε := ρε ∗ χE , where ρε is a standard mollifier. Note that fε is a smooth function belonging to W 1,1, (Rn ) and that fε → χE a.e. in Rn . Then, fix ϕ ∈ Cc1 (Rn , Rn ), ||ϕ||∞ ≤ 1. Using the definition of fε , performing a change of variable and recalling Definition 1.1, we easily get Z Z Z Z − ∇fε · ϕ dx = fε divϕ dx = dx ρε (z)χE (x − z)divϕ(x)dz n Rn Rn Rn ZR Z Z = ρε (z)dz χE (y)divϕ(y + z)dy ≤ P (E) ρε (z)dz = P (E). Rn

Rn

Rn

Taking the supremum over such ϕ we get Z P (E) ≥ Rn

n , |∇fε | ≥ C0 kfε k n−1

from which (1) follows, letting ε → 0 and recalling that fε (x) → χE (x) for a.e. x ∈ Rn . To show (1) ⇒ (2) we are going to use the Coarea formula (7). Note that by density it is enough to prove (2) for a function f ∈ Cc1 (Rn ). Moreover, splitting f in its positive and negative part, we may always assume without loss of generality that f ≥ 0. Then, for any τ ≥ 0 denote by fτ := min{f, τ }, the n . Note that φ is an increasing function and that truncation of f at the level τ . Set also φ(τ ) := kfτ k n−1 for h > 0 n φ(τ + h) − φ(τ ) ≤ kfτ +h − fτ k n−1 ≤ h|{f > τ }|1−1/n

10

Thus φ is Lipschitz and φ0 (τ ) ≤ |{f > τ }|1−1/n for a.e. τ ∈ R. Furthermore, using the isoperimetric inequality (5), we have Z +∞ Z +∞ n kf k n−1 |{f > s}|1−1/n ds φ0 (s)ds ≤ = lim φ(τ ) = τ →+∞ 0 0 Z Z +∞ Z +∞ Z −1 −1 −1 n−1 dH = C0 |∇f |, ≤ C0 P ({f > s})ds = C0 ds ∂ ∗ {f >s}

−∞

0

Rn

where the last equality follows from (7) with g ≡ 1.



2.1. Coarea formula for sets of finite perimeter. We conclude by recalling the coarea formula for sets of finite perimeter. Let f : Rn → Rk be a C 1 map, 1 ≤ k ≤ n − 1, and E a set of finite perimeter. By the definition of reduced boundary given in the first lecture, at every x ∈ ∂ ∗ E we have a generalized ∗ exterior normal ν E (x), hence a generalized tangent plane, which we will denote by Tx∂ E . Hence we can give the following Definition 2.8. If x ∈ ∂ ∗ E the tangential differential of f at x is the map df (x) : Tx∂ ∀ τ ∈ Tx∂

df (x)(τ ) = ∇f (x)(τ ),

∗E

∗E

→ Rk given by

.

Furthermore, we define the coarea factor at x as q Ck df (x) = det(df (x) ◦ (df (x))T ), where (df (x))T is the transpose of the matrix df (x). It can be shown that Ck df (x) is the square root of the sum of the squares of the k–order minors of ∗ the matrix representing df (x) with respect to a base in Tx∂ E and a base in Rk (see [2, (2.71)]). Theorem 2.9 (Coarea formula for sets of finite perimeter). If g : Rn → [0, +∞] is a Borel function, then Z Z Z g(x)Ck df (x)dHn−1 (x) = dτ g(x)dHn−1−k (x). ∂∗E

f −1 (τ )∩∂ ∗ E

Rk

Exercise 2.10. Prove that if f : Rn → Rk is the projection over the first k components, i.e. f (x, y) = x for all (x, y) ∈ Rk × Rn−k , then Ck df (x, y) = |νyE (x, y)|, for all (x, y) ∈ ∂ ∗ E, where ν E = (νxE , νyE ) ∈ Rk × Rn−k . In this case the coarea formula above becomes Z Z (8) g(x, y)|νyE (x, y)|dHn−1 (x, y) = ∂∗E

Rk

Z dx

g(x, y)dHn−1−k (y),

(∂ ∗ E)x

where (∂ ∗ E)x = {y ∈ Rn−k : (x, y) ∈ ∂ ∗ E}. In particular, recalling that H0 is the counting measure, if f is the projection over the first n − 1 components, we have that for every Borel function g : Rn → [0, +∞] Z Z  X  (9) g(x, y)|νyE (x, y)|dHn−1 (x, y) = g(x, y) dx. ∂∗E

Rn−1 {y∈(∂ ∗ E) } x

11

Figure 8. Section of ∂ ∗ E From formula (9) we may conclude that the vertical parts of ∂ ∗ E “are not seen from below”. To be precise, set V := {(x, y) ∈ ∂ ∗ E : νyE (x, y) = 0} and apply the formula with g = χV . Then R n−1 (x, y) = 0. Therefore, also the right hand side of (9) vanishes, i.e., E ∂ ∗ E χV (x, y)|νy (x, y)|dH

Figure 9. The set V Z # ({y : (x, y) ∈ V }) dx = 0. This implies that for Hn−1 -a.e. x ∈

Rn−1 Rn−1 ,

the section Vx is empty.

3. Third lecture: the quantitative isoperimetric inequality for convex sets After having proved the isoperimetric inequality we now turn to the next issue, namely the stability of this inequality. In other words, if E is a set such that |E| = |BR | and P (E) = P (BR ) + ε for some small ε, can we say that E is somehow close to a ball? and how can we measure the distance from a ball 12

Figure 10. Bonnesen’s Theorem in terms of ε? The first results in this direction were proven for planar convex sets by Bernstein [7] in 1905 and Bonnesen [8] in 1924. As we shall see in this lecture, it took some time before the problem was completely solved for convex sets in any dimension. Theorem 3.1 (Bonnesen). Given a convex set E ⊂ R2 , with |E| = |B1 |, there exist two concentric disks Br1 (x0 ) ⊂ E ⊂ Br2 (x0 ) such that P 2 (E) − P 2 (B1 ) . 4π A remarkable feature of this inequality is that the constant appearing on the right hand side is optimal. However, we cannot expect that such a precise estimate holds in higher dimension. Thus, it can be useful to restate it in a weaker form that we may hope to extend to the general n-dimensional case. To this aim, observe that from (10) it follows that if P (E) − P (B1 ) ≤ 1 there exists x0 ∈ R2 such that

(10)

(r2 − r1 )2 ≤

d2H (E, B1 (x0 )) ≤ C(P (E) − P (B1 )) for some positive constant C, where we denoted by dH (E, F ) := inf{ε > 0 : E ⊂ F + Bε , F ⊂ E + Bε } the Hausdorff distance between any two sets E, F ⊂ Rn . ¯ := {K ⊂ Ω ¯ : K compact}. Given Exercise 3.2. Let Ω ⊂ Rn be a bounded open set. Set C(Ω) ¯ Kj , K ∈ C(Ω), show that dH (Kj , K) → 0 if and only if (i) ∀ x ∈ K there exist xj ∈ Kj such that xj → x; (ii) if xj ∈ Kj , then any limit point of the sequence {xj } belongs to K. ¯ dH ) is a compact metric space. Moreover, (C(Ω), Definition 3.3. Let E ⊂ Rn be a convex set with |E| = |B1 |. We define the isoperimetric gap and the asymmetric index of E as D(E) := P (E) − P (B1 ), A(E) := minn dH (E, B1 (x)), x∈R

respectively. Using these quantities Fuglede in 1989 extended Bonnesen’s result to higher dimension, see [27]. 13

Figure 11. A nearly spherical set Theorem 3.4 (Fuglede). Let n ≥ 2. There exist δ, C, depending only on n, such that if E is convex, |E| = |B1 |, and D(E) ≤ δ, then:  p  n=2 C D(E),  s      1 A(E) ≤ C D(E) log , n=3  D(E)    2 C(D(E)) n+1 , n ≥ 4. Remark 3.5. As shown in [27], for n = 2 and n ≥ 4 the powers appearing in the previous estimates are optimal. Fuglede’s theorem will follow from the following result for “nearly spherical” sets. As we shall see in the fifth lecture this theorem will be also the starting point for a proof of the quantitative isoperimetric inequality for general sets of finite perimeter. Throughout all this lecture we set Σ := ∂B1 . Theorem 3.6. There exists ε(n) > 0 such that if E is an open set, |E| = |B1 |, the barycenter of E is the origin and (11)

∂E = {x + xu(x) : x ∈ Σ},

with kukW 1,∞ (Σ) < ε,

then (12)

D(E) ≥ Ckuk2W 1,2 (Σ) ≥ C0 |E∆B1 |2 ,

for some positive constants C, C0 , depending only on n. In the sequel we shall refer to a set E ⊂ Rn satisfying (11) as to a nearly spherical set. In order to prove Theorem 3.6 we need some formulas, whose proof is left as an exercise. Exercise 3.7. If E is a nearly spherical set, then s Z Z |∇τ u|2 1 n−1 n−1 P (E) = (1 + u) 1+ dH , |E| = (1 + u)n dHn−1 , (1 + u)2 n Σ Σ Z 1 b := barycenter = (1 + u(x))n+1 x dHn−1 (x), nωn Σ where ∇τ u denotes the tangential gradient of u. 14

Proof of Theorem 3.6. Let us estimate the isoperimetric gap, using the fact that kukW 1,∞ (Σ) ≤ ε. s # Z " |∇τ u|2 n−1 (1 + u) 1+ − 1 dHn−1 D(E) = P (E) − P (B1 ) = 2 (1 + u) Σ "s # Z Z 2   |∇ u| τ (1 + u)n−1 − 1 dHn−1 + (1 + u)n−1 = 1+ − 1 dHn−1 . (1 + u)2 Σ Σ Using Taylor expansion we find that   1 |∇τ u|2 1 |∇τ u|4 (1 + u)n−1 − dHn−1 2 4 2 (1 + u) 8 (1 + u) Σ Σ Z Z   1 n−1 n−1 ≥ (1 + u) − 1 dH + − cε |∇τ u|2 dHn−1 . 2 Σ Σ Z

D(E) ≥ (13)

  (1 + u)n−1 − 1 dHn−1 +

Z

Now, from the volume constraint |E| = |B1 | we get that Z   (1 + u)n − 1 dHn−1 = 0, Σ

that is n   X n h n−1 u dH = 0, nu + h Σ

Z (14)

h=2

which in turn gives Z Z n−1 2 n−1 udH ≥− u dH − cε u2 dHn−1 . 2 Σ Σ Σ Using this inequality we may estimate Z Z n−1 X n − 1 Z n−1 n−1 n−1 uh dHn−1 = [(1 + u) − 1]dH = (n − 1) udH + h Σ Σ Σ h=2 Z Z Z (n − 1)(n − 2) ≥ (n − 1) udHn−1 + u2 dHn−1 − cε u2 dHn−1 2 Σ Σ ZΣ Z n−1 2 n−1 2 n−1 ≥− u dH − cε u dH . 2 Σ Σ Z

n−1

In conclusion, recalling (13), we have proved that if kukW 1,∞ (Σ) ≤ ε, then 1 Z n − 1 Z 2 n−1 (15) D(E) ≥ − cε |∇τ u| dH − + cε u2 dHn−1 . 2 2 Σ Σ Now, for any integer k ≥ 0, let us denote by yk,i , i = 1, . . . , G(n, k), the spherical harmonics of order k, i.e. the restriction to Σ of the homogeneous harmonic polynomials of degreeRk, normalized so that √ ||yk,i ||L2 (Σ) = 1, for all k, i ∈ {1, . . . , G(n, k)}. For instance, y0 = 1/ nωn , since Σ 1dHn−1 = nωn , and R 2 n−1 R √ = n−1 Σ |x|2 dHn−1 = ωn . Then, since the functions y1,i = xi / ωn , for i = 1, . . . n, since Σ xi dH yk,i are eigenfunctions of the Laplace-Beltrami operator on Σ and −∆Σ yk,i = k(k + n − 2)yk,i , 15

we can write u=

∞ G(n,k) X X

Z ak,i yk,i ,

where

ak,i =

uyk,i dHn−1

Σ

k=0 i=1

and ||u||2L2 (Σ)

(16)

=

∞ G(n,k) X X

a2k,i ,

||∇τ u||2L2 (Σ)

=

k=0 i=1

∞ X

G(n,k)

k(k + n − 2)

X

a2k,i .

i=1

k=1

Observe that from formula (14) we have 1 a0 = √ nωn

n  Z X 1 n udHn−1 = − √ uh dHn−1 , n nω h n Σ Σ

Z

h=2

hence |a0 | ≤ c||u||22 ≤ cε||u||2 . Similarly, recalling that the barycenter of E is at the origin, i.e., Z (1 + u)n+1 xi dHn−1 = 0, for i = 1, . . . , n Σ

and that

R

Σ xi

= 0, we immediately get that for all i = 1, . . . , n, Z 1 n−1 ≤ cε||u||2 . uxi dH |a1,i | = √ ωn Σ

Therefore, from (16) we get ||u||22

2

≤ cε

||u||22

+

∞ G(n,k) X X

2

|ak,i | =⇒

||u||22

k=2 i=1

∞ G(n,k) 1 X X ≤ |ak,i |2 . 1 − cε k=2 i=1

But since for k ≥ 2, k(k + n − 2) ≥ 2n, from (16) we have ||u||22 ≤

(17)

1 ||∇τ u||22 2n(1 − cε)

and thus, recalling (15) and choosing ε sufficiently small, in dependence on n, we get 1 Z n − 1  1 D(E) ≥ − cε |∇τ u|2 dHn−1 − + cε ||∇τ u||22 2 2 2n(1 − cε) Σ Z ≥ c(n) |∇τ u|2 dHn−1 ≥ c(n)||u||2W 1,2 (Σ) , Σ

where the last inequality follows from (17).

 16

3.1. Proof of Fuglede’s Theorem 3.4. Before proving Fuglede’s Theorem 3.4 we indicate the main steps in the proof. Step 1: reduce the case of a general convex set to a convex set E, where kuk∞ < ε (this will be done by a contradiction argument). p 1 + kukL∞ Step 2: prove that if E is convex and kukL∞ < 1/2, then k∇τ ukL∞ ≤ 2 kukL∞ . 1 − kukL∞ Step 3: conclude using Theorem 3.6. We still need a couple of auxiliary results. Next one is proved in [27, Lemma 1.4] R Lemma 3.8 (Interpolation lemma). If v ∈ W 1,∞ (Σ) and Σ v = 0, then   π||∇τ v||2 , n=2    8e||∇ v|| τ ∞ 4||∇τ v||22 log ||v||n−1 2 , n=3 L∞ (Σ) ≤  ||∇ v|| τ  2  c(n)||∇ v||2 ||∇ v||n−3 , n ≥ 4, τ τ ∞ 2 where the constant c(n) depends only on the dimension. From Lemma 3.8 and Theorem 3.6 we easily get the following improved estimate. Theorem 3.9. Under the assumptions of Theorem 3.6, there exist ε, C > 0 depending only on n such that if ||u||W 1,∞ (Σ) ≤ ε, then  p C D(E),     n=2  1 ||u||n−1 L∞ (Σ) ≤ CD(E) log D(E) , n = 3   CD(E)||∇τ u||n−3 n ≥ 4. ∞ , Proof. Set v :=

Moreover, since

(1 + u)n − 1 . From the volume constraint |E| = |B1 | we have n Z Z   1 v dHn−1 = (1 + u)n − 1 dHn−1 = 0. n Σ Σ n   1X n h v =u+ u , n h h=2

if ε > 0 is small enough we have 1 1 |u| ≤ |v| ≤ 2|v|, |∇τ u| ≤ |∇τ v| ≤ 2|∇τ u|. 2 2 Then the result follows immediately from Theorem 3.6 and the interpolation Lemma 3.8.



The proof of Theorem 3.4 is now a quite simple consequence of the next two lemmas whose proof will be given later. The first lemma correspond to the Step 1 indicated above. Lemma 3.10. For all ε > 0, there exists δε > 0 such that if E is convex, |E| = |B1 |, the barycenter of E is the origin and D(E) < δε , then ∂E = {x(1 + u(x)) : x ∈ Σ}, The following lemma is the content of Step 2 above. 17

with ||u||L∞ (Σ) ≤ ε.

Lemma 3.11. If E is a convex set such that ∂E = {x(1 + u(x)) : x ∈ Σ}

1 and ||u||L∞ (Σ) ≤ , 2

then p 1 + kukL∞ . k∇τ ukL∞ ≤ 2 kukL∞ 1 − kukL∞ Proof of Theorem 3.4. Let us prove the Theorem for n ≥ 4, since otherwise the proof is similar and even easier. From Lemmas 3.10 and 3.11 it follows that if E is a convex set with |E| = |B1 | and D(E) is sufficiently small, then, up to a translation, E is a nearly spherical set with barycenter at the origin, satisfying kukW 1,∞ < ε, where ε > 0 is the one provided by Theorem 3.9. Therefore, using this theorem and Lemma 3.11 again, we get n−3

n−3 2 ||u||n−1 ∞ ≤ cD(E)||∇τ u||∞ ≤ c||u||∞ D(E), n+1

hence, ||u||∞2 ≤ cD(E). Thus we may conclude that 2

A(E) ≤ dH (E, B1 ) = ||u||∞ ≤ c [D(E)] n+1 .  We still need to prove Lemma 3.10 and Lemma 3.11. The proof of Lemma 3.10 is based on the following simple inequality. Lemma 3.12. Let E ⊂ Rn be a bounded open convex set. Then diam(E) ≤ c(n)

[P (E)]n−1 . |E|n−2

Proof of Lemma 3.10. We argue by contradiction. Assume that there exist ε0 > 0 and a sequence of closed convex sets Ej such that |Ej | = |B1 |, the barycenter of Ej is the origin, D(Ej ) → 0, but ||uj ||L∞ (Σ) ≥ ε0 , where uj is the function whose graph on Σ coincides with the boundary of Ej . From Lemma 3.12 the sets Ej are equibounded and so by Exercise 3.2 we may assume that they converge in the Hausdorff distance to a closed set E. Note that E is convex and that the sequence Ej converge to E also in measure. Moreover, using Exercise 3.14, it is easily checked that P (Ej ) → P (E). But since D(Ej ) → 0, then P (E) = limj P (Ej ) = P (B1 ). Thus E is a ball, actually the unit ball centered at the origin, since all Ej have barycenter at the origin. This gives a contradiction, since Ej is converging in the Hausdorff sense to the unit ball B1 , while ||uj ||L∞ (Σ) ≥ ε0 for all j.  Exercise 3.13. Let E be an open set (not necessarily convex) such that ∂E = {z(1 + u(z)) : z ∈ Σ}, for some Lipschitz function u such that kukL∞ (Σ) < 1/2. Show that for Hn−1 -a.e. z ∈ Σ the exterior normal to E at z(1 + u(z)) ∈ ∂E is given by z(1 + u(z)) − ∇τ u(z) νE = p . (1 + u(z))2 + |∇τ u(z)|2 18

Proof of Lemma 3.11. By Exercise 3.13, since z · ∇τ u(z) = 0 for Hn−1 -a.e. z ∈ Σ, we have 1 + u(z) z · ν E (Pz ) = p , (1 + u(z))2 + |∇τ u(z)|2 where Pz = z(1 + u(z)). Then, denoting by H the projection of the origin on the tangent plane to E at OH Pz , we have (see Figure 12 below) OP = z · ν E (Pz ). Observe that z

OPz ≤ 1 + ||u||∞ ,

OH ≥ 1 − ||u||∞ .

Thus, 1 − ||u||∞ 1 + u(z) ≤ z · ν E (Pz ) = p , 1 + ||u||∞ (1 + u(z))2 + |∇τ u(z)|2 from which we get |∇τ u(z)|2 ≤ (1 + u(z))2



1 + ||u||∞ 1 − ||u||∞

2 −1=

4||u||∞ , (1 − ||u||∞ )2

thus concluding

Figure 12. The situation in the proof of Lemma 3.11 . 

2

|∇τ u(z)| ≤ 4||u||∞

1 + ||u||∞ 1 − ||u||∞

2 ,

whence the assertion follows.



Exercise 3.14. Let E, F be convex sets. Show that if E $ F , then P (E) < P (F ). Proof of Lemma 3.12. First observe that if n = 2 we trivially have diam(E) ≤ 12 P (E). So assume n ≥ 3. Let x, y ∈ ∂E such that diam(E) := d = |x − y|. Then, rotate and translate E so to reduce to the situation shown in Figure 13 below. Rd By Fubini’s Theorem, |E| = 0 Hn−1 (Et )dt, where Et = E ∩ {xn = t}. Then observe that there exists s ∈ (0, d) such that Hn−1 (Es ) ≥ |E|/d. Note that we may always assume that 0 < s ≤ d/2 (otherwise 19

Figure 13. The situation in the proof of Lemma 3.12 we just rotate E upside down). Let C be the cone in Figure 13 with base Es and vertex x. We have by Exercise 3.14 and coarea formula (8) Z d Z Z d 1 n−1 n−2 P (E) ≥ P (C) ≥ H (∂C \ Es ) = dt dH ≥ Hn−2 (∂Ct )dt C s ∂Ct |νt | s Z d d − t n−2 n−2 d Hn−2 (∂Es ) (d − s)Hn−2 (∂Es ) = ≥ . H (∂Es )dt = d−s n−1 2 n−1 s Using the isoperimetric inequality (6) we have H

n−2

n−2

(∂Es ) ≥ (n −

 1/(n−1)  n−1 1)ωn−1 H (Es ) n−1

≥ (n −

1/(n−1) 1)ωn−1



|E| d

 n−2 n−1

.

Thus,  P (E) ≥ c(n)d

|E| d

 n−2 n−1

,

whence the result follows.



3.2. The case of sets of finite perimeter. We now discuss the quantitative isoperimetric inequality for general sets of finite perimeter. In this case it is clear that we cannot use the Hausdorff distance to measure the asymmetry of a set and we have to replace it with an L1 -type distance. To this aim we give the following Definition 3.15. Given a measurable set E with |E| = |BR |, we denote by n |E∆B (x)| o P (E) − P (BR ) R α(E) := minn , D(E) = , n x∈R R Rn−1 the asymmetric index and the isoperimetric gap of E, respectively. We will refer to a ball minimizing α(E) as to an optimal ball. 20

In 1992 Hall ([35]), using some previous results proved in collaboration with by Hayman and Weitsman ([36]), showed that there exists a constant c(n) such that for all measurable sets of finite measure (18)

α(E) ≤ c(n)(D(E))1/4 .

It is interesting to observe that the power 1/4 appearing on the right hand side of this estimate does not depend on the dimension, while in Fuglede’s result it does. The reason is due to the fact that Fuglede uses the Hausdorff distance to measure the asymmetry, while Hall uses the L1 distance. However, Hall in his paper conjectures that the right power should be 1/2 and actually proves that (18) holds with 1/2 if E is an axially symmetric set. Moreover he observes that one cannot expect a power higher than 1/2. To see this, take ε > 0 and consider in any dimension n ≥ 2 the ellipsoid   2 x1 2 2 2 + x2 (1 + ε) + x3 + . . . + xn ≤ 1 . Eε = 1+ε It can be proved (see for instance [3, Lemma 5.9]) that α(Eε ) = |Eε ∆B1 |. Then one can show that D(Eε ) → γ > 0, as ε → 0+ . α2 (Eε ) Another way to convince oneself that the right exponent in the inequality (18) is 1/2 is to consider a set E obtained from a ball by adding and removing a small cone as illustrated in the Figure below.
If r is the √ radius of the base of the cone and ε δ0 , then |E∆BR | 2|E| 2ωn p √ α(E) ≤ D(E). ≤ = 2ω ≤ n Rn Rn δ0 The strategy of the proof consists in reducing the general case of a set of finite perimeter to smaller classes of sets. The first reduction is provided by the next result (see [32, Lemma 5.1]). Lemma 3.17. There exist positive constants L, C, δ depending only on n such that if |E| = |B1 | and D(E) ≤ δ one can find a set F ⊂ [−L, L]n , with |F | = |B1 |, such that α(E) ≤ α(F ) + CD(F ) and D(F ) ≤ CD(E). We will not prove Lemma 3.17 whose proof consists in cutting the far away parts of E and rescaling the remaining part of the set. The proof uses the classical isoperimetric inequality and the strict concavity of n−1 the function t n for t > 0, which allows to estimate in a quantitative way the isoperimetric gap created by splitting a set in two parts. To understand this observe that for all λ ∈ (0, 1) λ

n−1 n

+ (1 − λ)

n−1 n

− 1 ≥ c(n) min{λ, 1 − λ}.

Therefore, if Br (x) and Bρ (y) are two disjoint balls, with |x − y| >> 1, such that |Br (x)| + |Bρ (y)| = |B1 |, i.e., rn + ρn = 1, we may estimate the perimeter deficit of the set E = Br (x) ∪ Bρ (y) D(E) = P (Br (x)) + P (Bρ (y)) − P (B) ≥ c(n) min{rn , ρn } ≥ c(n)α(E). Note that if Theorem 3.16 is true for bounded sets, then using Lemma 3.17, it holds true for all sets with D(E) ≤ δ, hence for all sets, as observed before. As a matter of fact, if Theorem 3.16 holds for F we have hp i p p α(E) ≤ α(F ) + CD(F ) ≤ κ(n) D(F ) + CD(F ) ≤ c D(E) + D(E) ≤ c D(E), where C is the constant provided by Lemma 3.17. 4. Fourth lecture: the proof of the quantitative isoperimetric inequality via symmetrization In this lecture we are going to prove the quantitative isoperimetric inequality p α(E) ≤ κ(n) D(E) following, with some simplifications, the original argument in [32] which is mostly based on symmetrization procedures. In view of the preliminary remarks made in the previous lecture it is clear that in order to prove this inequality we may assume without loss of generality that the set E has volume ωn , that E ⊂ [−L, L]n , where L is the constant provided by Lemma 3.17, and that D(E) ≤ δ for some conveniently small δ. The advantage of working with bounded sets is that from the compactness theorem for sets of finite perimeter Theorem 1.16, one immediately gets that α(E) depends continuously on D(E). 22

Figure 15. The case of optimal ball not centered at the origin Exercise 4.1. For all ε > 0, there exists δε > 0 such that if E ⊂ [−L, L]n , |E| = |B1 |, δ(E) ≤ δε , then α(E) ≤ ε. Next step in the proof of the quantitative isoperimetric inequality is to reduce to the simpler case of an n–symmetric set. Definition 4.2. We say that E ⊂ Rn is n–symmetric set if, up to a translation, E is symmetric about each coordinate plane. Note that even if E is n–symmetric it is not true in general that the optimal ball is the one centered at the center of symmetry of E, as shown in Figure 15 below. However, next lemma shows that for n–symmetric sets, this ball is optimal “up to a constant”. Lemma 4.3. Let E be n–symmetric with centre of symmetry at the origin, |E| = |B1 |. Then α(E) ≤ |E∆B1 | ≤ 3α(E) Proof. Let B1 (x0 ) be an optimal ball for E, i.e. α(E) = |E∆B1 (x0 )|. Then by the triangular inequality we have |E∆B1 | ≤ |E∆B1 (x0 )| + |B1 (x0 )∆B1 |. Note that since E is n–symmetric, then B1 (−x0 ) is optimal as well, i.e. α(E) = |E∆B1 (−x0 )|. Therefore from the inequality above we have α(E) ≤ |E∆B1 | ≤ α(E) + |B1 (x0 )∆B1 | ≤ α(E) + |B1 (x0 )∆B1 (−x0 )| ≤ α(E) + |E∆B1 (x0 )| + |E∆B1 (−x0 )| ≤ 3α(E).  The next result will allow us to reduce the proof of the quatitative isoperimetric inequality (19) to n–symmetric bounded sets. Theorem 4.4. There exist δ1 and C1 depending only on n such that if E ⊂ [−L, L]n , |E| = |B1 |, δ(E) ≤ δ1 , then there exists an n–symmetric set F such that F ⊂ [−L, L]n , |F | = |B1 | and D(F ) ≤ 2n D(E).

α(E) ≤ C1 α(F ), 23

Figure 16. Construction of E + and E − The proof of Theorem 4.4 is quite tricky. In order to explain the strategy we give some simple definitions. Given a direction ν ∈ Sn−1 and a set E, let us consider the affine hyperplane πν orthogonal to ν splitting E into two parts of equal measure. We denote by E 0 the part of E contained in the half space with inner normal ν and by E 00 the complement of E 0 in E. Then, we set Eν+ := E 0 ∪ rν (E 0 ), where rν is the reflection about the hyperplane πν and Eν− := E 00 ∪ rν (E 00 ). The construction is illustrated in Figure 16 below where, to simplify the notation, we dropped the subscript ν. Exercise 4.5. Given a set E of finite perimeter and a direction ν, show that P (Eν+ ) + P (Eν− ) ≤ 2P (E) and thus D(Eν± ) ≤ 2D(E). Give an example showing that the inequality may be strict. Now, since D(Eν+ ) ≤ 2D(E) and D(Eν− ) ≤ 2D(E), if for some universal constant c(n) we had (20)

α(E) ≤ c(n)α(Eν+ )

or α(E) ≤ c(n)α(Eν− ),

then the proof of the quantitative isoperimetric inequality for a bounded set would be reduced to the case of a set with one plane of symmetry. Then, symmetrizing with respect to all possible n orthogonal directions, we would reduce to the case of n–symmetric sets. Unfortunately, (20) is false, as shown by the example in Figure 17, where we have α(E) > 0, but α(Eν+ ) = α(Eν− ) = 0.

Figure 17. Counterexample to (20) The following lemma (see [32, Lemma 2.5]) deals with the situation that occurs in this example. Lemma 4.6. There exists δ, C, depending only on n, such that if E ⊂ [−L, L]n , |E| = |B1 | and D(E) ≤ δ, given any two orthogonal direction ν1 , ν2 and the four sets Eν+1 , Eν−1 , Eν+2 , Eν−2 , we have that D(Eν±i ) ≤ 2D(E), for i = 1, 2. Moreover, at least one of them, call it F , satisfies the estimate α(E) ≤ Cα(F ). 24

In other words, it may happen that some of the sets Eν±i have a small asymmetry (or even zero asymmetry as in Figure 17), but at least one of them must have asymmetry larger than E up to a constant. Note that if we symmetrize the set from the previous example in the horizontal direction we get a greater asymmetry, see Figure 18.

Figure 18. A different symmetrization may give a bigger asymmetry We are now ready to prove Theorem 4.4. Proof of Theorem 4.4. Take δ1 = 2−(n−1) δ, where δ is the constant appearing in Lemma 4.6 above. By ˜ ⊂ [−L, L]n applying the lemma n − 1 times to different pairs of orthogonal directions we find a set E ˜ with n − 1 symmetries, |E| = |B1 |, such that ˜ α(E) ≤ C n−1 α(E),

˜ ≤ 2n−1 D(E), D(E)

˜ is where C is the constant given by Lemma 4.6. Without loss of generality we may assume that E symmetric with respect to the first n − 1 directions e1 , . . . , en−1 . Let us consider a hyperplane πen ˜ into two parts of equal measure, E ˜ 0, E ˜ 00 and the corresponding sets E ˜± . orthogonal to en and dividing E en From Exercise 4.5 we have that ˜ ± ) ≤ 2D(E) ˜ ≤ 2n D(E). D(E en ˜e± observe that since E ˜ is symmetric with respect to the first n − 1 To control the asymmetry of E n ± ˜ ˜ if necessary, we may directions, the sets Een are both n–symmetric. Moreover, by suitably translating E also assume that they are both symmetric around the origin. Thus we may apply Lemma 4.3 to estimate − + ˜ ≤ |E∆B ˜ ˜0 ˜ 00 α(E) 1 | = |E ∆B1 | + |E ∆B1 |  3  1  ˜+ ˜e+ ) + α(E ˜− ˜e− ) . α(E = |E en ∆B1 | + |Een ∆B1 | ≤ n n 2 2 ˜ Therefore, denoting by F ˜e± has asymmetry index greater than 1 α(E). Thus, at least one of the sets E n 3 this set, we have ˜ ≤ 2n D(E) D(F ) ≤ 2D(E)

and ˜ ≤ 3C n−1 α(F ). α(E) ≤ C n−1 α(E)  Having proved Theorem 4.4, from now on we may assume that E is an n–symmetric set such that E ⊂ [−L, L]n , |E| = |B1 |. We now want to pass from n–symmetric sets to axially symmetric sets, i.e., sets E having an axis of symmetry such that every non-empty cross-section of E perpendicular to this axis is a (n − 1)-dimensional ball. 25

Figure 19. Schwartz symmetral of the set E In order to perform this further simplification, let us recall the definition of Schwartz symmetrization of a set E. To this aim, given a measurable set E, for all t ∈ R we set Et = {x ∈ Rn−1 : (x, t) ∈ E}. Note that if E is a set of finite perimeter, then by Vol’pert’s theorem Et is a set of finite perimeter in Rn−1 for a.e. t ∈ R. A proof of this result is given in [3, Th. 2.4]. Definition 4.7. Given a measurable set E ⊂ Rn , its Schwartz symmetrization is defined as E ∗ = {(x, t) ∈ Rn−1 × R : t ∈ R, |x| < rE (t)}, n−1 (t) = Hn−1 (Et ). where ωn−1 rE

Note that |E ∗ | = |E|. Moreover, as for Steiner symmetrization, also Schwartz symmetrization decreases the perimeter. Next result (see [32, Lemma 3.3]) contains some useful properties of the Schwartz symmetrization of a set of finite perimeter whose boundary has no horizontal flat parts. To this aim, given a measurable set E ⊂ Rn , for all t ∈ R we set vE (t) := Hn−1 (Et ),

pE (t) := Pn−1 (Et ), n−1 R . Note that from

where Pn−1 (·) denotes the perimeter of a subset of Definition 4.7 we have that vE (t) = vE ∗ (t) for all t. Moreover, the isoperimetric inequality in Rn−1 yields that pE ∗ (t) ≤ pE (t), since (E ∗ )t is a ball with the same measure of Et . Theorem 4.8. Let E ⊂ Rn be a set of finite perimeter such that Hn−1 (∂ ∗ E ∩ {νtE = ±1}) = 0.

(21)

Then the section Et is a set of finite perimeter in Rn−1 for a.e. t ∈ R, the function vE belongs to W 1,1 (R) and the following formulas hold: Z q Z q ∗ 02 2 02 + p2 dt. P (E ) = vE + pE ∗ dt, P (E) ≥ vE E R

R

The next step in the proof of the quantitative isoperimetric inequality is given by the following theorem, which states that we may eventually reduce to the case of axially symmetric sets. 26

Theorem 4.9. Let E ⊂ [−L, L]n be an n–symmetric set satisfying (21) such that |E| = |B1 | and D(E) ≤ 1. If n = 2 or if n ≥ 3 and the quantitative isoperimetric inequality (19) holds true in Rn−1 , there exists a constant C depending only on n such that p (22) α(E) ≤ α(E ∗ ) + C D(E), and D(E ∗ ) ≤ D(E). To conclude the proof of the quantitative isoperimetric inequality (19) we have to combine this result with a final estimate for axially symmetric sets. This is provided by the next theorem, which is a particular case of a more general proved by Hall in [35, Th. 2] for axially symmetric sets. The proof of estimate (23) requires some long, but one-dimensional calculations (compare also with a slightly weaker result given in [32, Th. 4.1]). Theorem 4.10. Let E ⊂ [−L, L]n be an axially and n-symmetric set with center of symmetry at the origin, such that |E| = |B1 |. Then p (23) |E∆B1 | ≤ C 0 D(E), for some constant C 0 depending only on the dimension n. Let us now show how the two previous theorems lead at once to the proof of Theorem 3.16. Then we will prove Theorem 4.9. Proof of the quantitative isoperimetric inequality (19). We argue by induction on the dimension n assuming that either n = 2 or n ≥ 3 and the isoperimetric inequality (19) holds in Rn−1 . As we have already observed, in order to prove (19) for a set of finite perimeter E ⊂ Rn , we may assume without loss of generality that E is an n–symmetric set contained in [−L, L]n , that |E| = |B1 | and that D(E) ≤ 1. Morover, by rotating and translating E if necessary, we may also assume that Hn−1 (∂ ∗ E ∩ {νtE = ±1}) = 0 and that E is symmetric with respect to the origin. In particular E satisfies all the assumptions of Theorem 4.9, while E ∗ satisfies the assumptions of Theorem 4.10. Thus, recalling (22) and (23) we conclude that p p α(E) ≤ α(E ∗ ) + C D(E) ≤ |E ∗ ∆B1 | + C D(E) p p p ≤ C 0 D(E ∗ ) + C D(E) ≤ c(n) D(E), where c(n) is the sum of the two constants C and C 0 appearing in (22) and (23), respectively.



We now turn to the proof of Theorem 4.9. Proof of Theorem 4.9. Denoting by B1 (x0 ) an optimal ball for E ∗ , we have α(E) ≤ |E∆B1 (x0 )| ≤ |E ∗ ∆B1 (x0 )| + |E∆E ∗ | = α(E ∗ ) + |E∆E ∗ |. Hence, in order to prove the first inequality in (22) it is enough to show that p (24) |E∆E ∗ | ≤ c(n) D(E), for some positive constant c depending only on n. The second inequality in (22) follows immediately from Theorem 4.8 which yields that P (E ∗ ) ≤ P (E), hence D(E ∗ ) ≤ D(E). 27

To prove (24) we use again Theorem 4.8 to estimate Z q q 02 02 + p2 2 D(E) = P (E) − P (B1 ) ≥ P (E) − P (E ) ≥ vE + p E − vE E∗ R Z p2E − p2E ∗ q q = 02 + p2 + 02 + p2 R vE vE E E∗ 2 Z q 1 q p2E − p2E ∗ ≥ R q 02 2 + 02 + p2 R v + p vE E E E∗ R Z q 2 1 ≥ p2E − p2E ∗ , P (E) + P (E ∗ ) R ∗

where the inequality before the last one follows from H¨older’s inequality. Since D(E) ≤ 1, we have P (E ∗ ) ≤ P (E) ≤ P (B1 ) + 1. Therefore from the above estimate we get, recalling that pE ≥ pE ∗ , Z q p (25) D(E) ≥ cn p2E − p2E ∗ R Z Z p p √ √ √ pE + pE ∗ pE ∗ (pE − pE ∗ )/pE ∗ ≥ 2cn pE ∗ (pE − pE ∗ )/pE ∗ . = cn R

R

(E ∗ )

Now assume that n ≥ 3 and observe that since t is a (n − 1)–dimensional ball of radius rE (t) with n−1 (t) = Hn−1 (Et ), the ratio ωn−1 rE pE (t) − pE ∗ (t) n−2 (t) rE is the isoperimetric gap in Rn−1 of Et . Since by assumption, the quantitative isoperimetric inequality (19) holds true in Rn−1 , we have s pE (t) − pE ∗ (t) κ(n − 1) ≥ αn−1 (Et ), n−2 (t) rE where αn−1 (Et ) is the (n − 1)- dimensional asymmetry of Et . But Et is an (n − 1)–symmetric set in Rn−1 and (E ∗ )t is the ball centered at the center of symmetry of Et . Therefore from Lemma 4.3 we get s pE (t) − pE ∗ (t) 1 Hn−1 (Et ∆(E ∗ )t ) κ(n − 1) ≥ α (E ) ≥ . n−1 t n−2 n−1 3 rE (t) rE (t) Inserting this inequality in (25) we then get s Z Z p pE (t) − pE ∗ (t) Hn−1 (Et ∆Et∗ ) n−2 D(E) ≥ c rE (t) dt ≥ c n−2 rE (t) rE (t) R R Z L c c ≥ Hn−1 (Et ∆Et∗ ) = |Et ∆Et∗ |, L −L L where the inequality before the last one follows from the inclusion E ⊂ [−L, L]n and the last equality is just Fubini’s theorem. This proves (24). Hence the assertion follows when n ≥ 3. If n = 2, since E is 2-symmetric, either Et is a symmetric interval (and thus Et = Et∗ ) or Et is the union 28

of at least two essentially disjoint intervals and thus pE (t) ≥ 4, while pE ∗ (t) = 2. Note also that since E ⊂ [−L, L]2 , then H1 (Et ∆Et∗ ) ≤ 2L for all t ∈ R. Therefore, from (25) we easily get Z Z p p √ √ pE ∗ (pE − pE ∗ )/pE ∗ dt = 2c2 D(E) ≥ 2c2 pE − pE ∗ dt R

Z ≥ 2c2

√

√ 2 dt ≥

{t: Et 6=Et∗ }

2c2 L

{t: Et 6=Et∗ }

Z H {t: Et 6=Et∗ }

1

(Et ∆Et∗ )dt

√ =

2c2 |E∆E ∗ |, L

thus concluding the proof also in this case.



5. Fifth lecture: alternative proofs of the quantitative isoperimetric inequality 5.1. Isoperimetric inequalities via Optimal Mass Transport. In the last lecture we presented the proof of the the quantitative isoperimetric inequality given in [32]. As we have seen, that proof was based on symmetrization arguments. The same approach has been used in several other papers to obtain quantitative versions of the Sobolev inequality, of the isoperimetric inequality in Gauss space and of other relevant geometric and functional inequalities (see for instance [31], [13], [21], [22], [33], [14], [34], [10], [3], [24], [4], and also [29], [45], [40]). On the other hand there are situations where one considers inequalities which are realized by non symmetric sets or functions. This is the case of the anisotropic isoperimetric inequality. In order to state it let us fix some notation. Let Γ : Rn → [0, ∞) be a positively 1-homogeneous function such that Γ(x) > 0 for all x 6= 0. To the function Γ we may associate the anisotropic perimeter, defined for any set E of finite perimeter by setting Z PΓ (E) := Γ(ν E (x))dHn−1 (x). ∂∗E

It is well known that the isoperimetric sets with respect to this perimeter are the homothetic and translated of the so called Wulff shape set associated to Γ which is given by WΓ := {x ∈ Rn : hx, νi − Γ(ν) < 0 for all ν ∈ Sn−1 }. Then, the anisotropic isoperimetric inequality states that PΓ (E) ≥ PΓ (WΓ ) for all sets of finite perimeter such that |E| = |WΓ |, with equality holding if and only if E is a translated of the Wulff shape set WΓ (see [26] and also [17] for a simpler proof in two dimensions). The quantitative version of the anisotropic isoperimetric inequality is a remarkable result proved by Figalli, Maggi and Pratelli in [23]. It states that there exists a constant C, depending only on n and Γ, such that for any set of finite perimeter E such that |E| = rn |WΓ | p (26) αΓ (E) ≤ C DΓ (E), where

n |E∆(x + rW ) o PΓ (E) − PΓ (rWΓ ) Γ , DΓ (E) := . n x∈R r rn−1 denote the anisotropic asymmetry index and the anisotropic isoperimetric gap, respectively. Since the Wulff shape WΓ can be any bounded open convex set, it is clear that no symmetrization argument can be used to prove the anisotropic isoperimetric inequality or its quantitative counterpart (26). And in fact the strategy used in [23] is completely different from the one we have seen in the last αΓ (E) := minn

29

Figure 20. The Brenier’s map lecture, since it relies on an optimal mass transportation argument and on the proof of a very general trace inequality. To simplify even further the presentation of the main ideas used in the proof of Figalli, Maggi and Pratelli we shall only consider the case of the standard perimeter where inequality (26) reduces to the more familiar quantitative isoperimetric inequality (19). The starting point is a variant of the Gromov’s proof of the classical isoperimetric inequality where the Knothe map originally used in [43] is replaced by the Brenier map whose existence is stated in the following result (see [11] and also [41], [42]). Theorem 5.1. Let E be a set of finite perimeter with |E| = |B1 |. There exists a convex function ϕ : Rn → R such that if we set T = ∇ϕ, then T (x) ∈ B1 for a.e. x ∈ Rn and det ∇T (x) = 1 for a.e. x ∈ E. Note that there is a regularity issue here, since if ϕ is convex T is defined only a.e. and it is just a BV map. However, let us assume that E is sufficiently smooth to ensure that T is a Lipschitz map. Under this extra assumption we may now give the Gromov’s proof of the isoperimetric inequality. For every x ∈ E denote by λi (x), i = 1, . . . , n, the eingenvalues of the symmetric matrix ∇T (x). Using the arithmetic–geometric mean inequality, we have Z Z Z 1/n P (B1 ) = nωn = n dy = n (det ∇T ) dx = n (λ1 . . . λn )1/n dx B1 E E Z Z Z ≤ (λ1 + · · · + λn )dx = div T dx = T · ν E dHn−1 ≤ P (E). E

E

∂E

Observe that if P (E) = P (B1 ) then λ1 (x) = λ2 (x) = . . . = λn (x) = 1 for a.e. x ∈ E, since det ∇T (x) = 1. Hence, up to a translation, T is the identity map and E is a ball.  Let us try to exploit this argument to prove the quantitative isoperimetric inequality. Since, by definition, P (E) = P (B1 ) + D(E), from the inequalities above we immediately get that Z h i 1 (27) (λ1 + . . . λn )/n − (λ1 . . . λn )1/n ≤ D(E), n E Z (28) (1 − T · ν E )dHn−1 ≤ D(E). ∂E 30

It is not too difficult to show (see [23, Corollary 2.4]) that (27) implies that there exists a constant c depending only on n such that Z p |∇T − I| ≤ c D(E), (29) E

where I stands for the identity matrix, provided D(E) ≤ 1. Let us assume for a moment that the set E is so good to satisfy a Poincar´e inequality for some universal constant depending only on n and let us see what information we may deduce from (29). We have Z Z p |∇T − I| ≤ c D(E). |T x − x| ≤ C E

E

By translating E we may assume that its optimal ball is the unit ball B1 . Therefore, given ε > 0, from the previous inequality we get, for ε > 0, α(E) = |E∆B1 | = 2|E \ B1 | ≤ 2(|E \ B1+ε | + |B1+ε \ B1 |) Z p −1 ≤ C(ε + ε |T x − x|) ≤ C(ε + ε−1 D(E)). E

Minimizing the right hand side of this inequality with respect to ε we then get α(E) ≤ c(n)(D(E))1/4 , that is the quantitative isoperimetric inequality with the not optimal exponent 1/4. Note however that this argument can never lead to a proof of the quantitative isoperimetric inequality: firstly, because even if E is a connected open set the constant of the Poincar´e inequality may blow up in presence of small cusps; secondly because in the above argument we are not taking into a account the information contained in the inequality (28) derived from Gromov’s proof. Indeed the strategy followed in [23] is more subtle. Namely, one can show that if E has small deficit, then (see [23, Th. 3.4 and Lemma 3.1]), up to removing a small critical set from E (whose measure is controlled by |E|D(E)), one may assume that E satisfies the following trace inequality Z Z n−1 inf |f − c| dH ≤ τ (n) |∇f | dx for all f ∈ C 1 (Rn ), c

∂∗E

E

for some constant τ depending only on n. Then, if we apply the previous inequality to the map T − Id, up to translating E conveniently, we have, recalling (29), Z Z p |T x − x| dHn−1 ≤ τ (n) |∇T − I| dx ≤ c D(E). ∂∗E

E

Combining this inequality with (28), we obtain Z Z   |1 − |x|| dHn−1 ≤ |1 − |T x|| + ||T x| − |x|| dHn−1 ∗ ∂∗E Z ∂ E i h p p 
 1 − T · ν E + |T x − x| dHn−1 ≤ c D(E) + D(E) ≤ c D(E). ∂∗E

The proof of the quantitative isoperimetric inequality (19) then immediately follows from this estimate, since by [23, Lemma 3.5]) there exists a constant c(n) such that (see the figure below) Z c(n)|E∆B1 | ≤ |1 − |x|| dHn−1 . ∂E 31

Figure 21. c|E∆B1 | ≤

R

∂∗E

|1 − |x||

Beside providing an alternative proof of the quantitative isoperimetric inequality in the wider framework of anisotropic perimeter, the paper by Figalli, Maggi and Pratelli contains several interesting results. In particular, Theorem 3.4 which states that given any set of finite perimeter E with small deficit one may always extract from E a maximal set for which a trace inequality holds with a universal constant. This is a new and deep result that may have several applications. Moreover, the mass transportation approach used in [23] has been also successfully used to obtain the quantitative versions of other important inequalities (see [13], [22], [24]). 5.2. Isoperimetric inequality via the regularity theory of minimal surfaces. Another very interesting proof of the quantitative isoperimetric inequality has been recently given by Cicalese and Leonardi in [15]. Their starting point is the quantitative inequality (12) proved by Fuglede for nearly spherical sets and the observation that all the known examples suggest that the quantitative inequality becomes really critical only when the set E is a small perturbation of a ball. Therefore their idea is to reduce the general case to the case of nearly spherical sets via a contradiction argument. Precisely, they start by assuming that there exists a sequence of sets Ej , converging in measure to the unit ball, for which the quantitative inequality does not hold. Then they replace them with a different sequence of sets Fj , still not satisfying the quantitative inequality, but converging to B1 in C 1 , thus contradicting Fuglede’s Theorem 3.6 for nearly spherical sets. The sets Fj are constructed as the solutions of certain minimum problems and their convergence in C 1 to the unit ball is a consequence of the a priori estimates for perimeter almost minimizers established in the theory of minimal surfaces. Though the approach of Cicalese and Leonardi to the quantitative isoperimetric inequality is based on the results of a difficult and deep theory, it has the advantage of providing a simple proof that has been successfully applied to several other inequalities (see [1], [5], [16], [6], [9]). The proof we are going to present here is a further simplification of the original proof by Cicalese and Leonardi which has been developed in a more general context by Acerbi, F. and Morini in [1]. Let us now quickly recall the definition and the regularity properties of the perimeter almost minimizers. Definition 5.2. Let ω, R be positive numbers. A set F of finite perimeter is an (ω, R)–almost minimizer if, for all balls Br (x0 ) with r < R and all measurable sets G such that F ∆G ⊂⊂ Br (x0 ), we have P (F ) ≤ P (G) + ωrn . 32

Figure 22. A perimeter almost minimizer F Thus, an almost minimizer locally minimizes the perimeter up a higher order error term. The main properties of almost minimizers are contained in the following statement which is essentially due to Tamanini (see [47, Sect. 1.9 and 1.10] and also [39, Th. 26.5 and 26.6]). Theorem 5.3. If E is an (ω, R)– almost minimizer, then ∂ ∗ E is a manifold of class C 1,1/2 , ∂E \ ∂ ∗ E is relatively closed in ∂E and H s (∂E \ ∂ ∗ E) = 0 for all s > n − 8. Moreover, if {Ej } is a sequence of equibounded (ω, R)– almost minimizers converging in measure to an open set E of class C 2 , then for j large each Ej is of class C 1,1/2 and the sequence {Ej } converge to E in C 1,α for all 0 < α < 1/2. As we said above the starting point of the proof is the Fuglede estimate for nearly spherical sets stated in Theorem 3.6. Recall that there exist two positive constants ε(n), C0 (n) such that if E is an open set with |E| = |B1 |, the barycenter of E is at the origin and ∂E = {x + u(x) : x ∈ ∂B1 } for a Lipschitz function u such that kukW 1,∞ < ε, then the following estimate holds D(E) ≥ C0 |E∆B1 |2 .

(30)

We also need the following simple lemma. Lemma 5.4. If Λ > n, the unique solution up to translations of the problem  (31) min P (F ) + Λ |F | − |B1 | : F ⊂ Rn is the unit ball. Proof. By the isoperimetric inequality it follows that in order to minimize the functional in (31), we may restrict to the balls Br . Thus the above problem becomes  min nωn rn−1 + Λωn |rn − 1|} , r>0

which has a unique minimum for r = 1, if Λ > n.



We are now ready to give the proof of the quantitative isoperimetric inequality (19) via regularity. Before that we need also to introduce the non-rescaled asymmetry index by setting for any measurable set of finite measure  A(E) := minn |E∆B1 (x)| . x∈R

Clearly A(E) = α(E) if |E| = |B1 |. 33

Proof of the quantitative isoperimetric inequality via regularity. Step 1. Fix R0 > 0 such that the ball BR0 contains the cube [−L, L]n provided by Lemma 3.17. As we have already observed in the previous lectures, in order to prove (19) it is enough to show that Claim. There exists δ0 > 0 such that if E ⊂ BR0 , |E| = |B1 | and D(E) ≤ δ0 , then C0 A(E)2 ≤ D(E), 2

(32)

where C0 is the constant appearing in (30). To this aim we argue by contradiction assuming that there exist a sequence Ej ⊂ BR0 , |Ej | = |B1 |, with D(Ej ) → 0 and (33)

P (Ej ) < P (B1 ) +

C0 A(Ej )2 . 2

Since D(Ej ) → 0, by the compactness Theorem 1.15 we may assume that up to a subsequence the sets Ej converge in measure to some set E. Then, by the lower semicontinuity of the perimeter we get that D(E) = 0. Thus E is a ball of radius 1 and we may conclude that A(Ej ) → 0 as j → ∞. Now, to achieve the proof of (32), we would like to replace {Ej } with a sequence of sets converging to B1 in C 1 and contradicting inequality (30). To build this new sequence, for every j we consider a minimizer Fj to the problem:  min P (F ) + |A(F ) − A(Ej )| + Λ||F | − ωn | : F ⊂ BR0 , where Λ > n is a fixed constant. Using again the compactness Theorem 1.15 we may assume that the sets Fj converge in measure to a set F . Moreover,the lower semicontinuity of the perimeter immediately yields that F is a minimizer of the problem: min P (E) + A(E) + Λ||E| − ωn | : E ⊂ BR0 . Therefore by Lemma 5.4 we may conclude that the sequence {Fj } converge in measure to a ball B1 (x0 ). Let us now show that this convergence holds indeed in C 1 . To this aim, by Theorem 5.3 it is enough to prove that each Fj is an (ω, R0 )–almost minimizer for some ω > 0. To prove this take a ball Br (x0 ) with r < R0 and a set G such that Fj ∆G ⊂⊂ Br (x0 ). Two cases may occur. ¯R . Then, by the minimality of Fj we get Case 1. G ⊂ B 0   P (Fj ) ≤ P (G) + |A(G) − A(Ej )| − |A(Fj ) − A(Ej )| + Λ ||G| − ωn | − ||Fj | − ωn | ≤ P (G) + |A(G) − A(Fj )| + Λ ||G| − |Fj || ≤ P (G) + (Λ + 1)|G∆Fj | ≤ P (G) + (Λ + 1)ωn rn . ¯R | > 0. In this case we split G as follows Case 2. |G \ B 0 P (Fj ) − P (G) = [P (Fj ) − P (G ∩ BR0 )] + [P (G ∩ BR0 ) − P (G)]. Since G ∩ BR0 ⊂ BR0 , as before we have P (Fj ) − P (G ∩ BR0 ) ≤ (Λ + 1)ωn rn , while P (G ∩ BR0 ) − P (G) = P (BR0 ) − P (G ∪ BR0 ) ≤ 0 34

Figure 23. E and F have the same measure and the same asymmetry by the isoperimetric inequality. Therefore we may conclude that the sets Fj are all ((Λ + 1)ωn , R0 )– almost minimizers and that the sequence {Fj } converges to B1 (x0 ) in C 1,α for all α < 1/2. Step 2. By the minimality of the Fj , recalling (33) and using Lemma 5.4, we get (34)

P (Fj ) + Λ||Fj | − ωn | + |A(Fj ) − A(Ej )| ≤ P (Ej ) C0 C0 A(Ej )2 ≤ P (Fj ) + Λ||Fj | − ωn | + A(Ej )2 . ≤ P (B1 ) + 2 2

Therefore, we have that |A(Fj )−A(Ej )| ≤ C20 A(Ej )2 and since A(Ej ) → 0, we get that A(Fj )/A(Ej ) → 1 as j → ∞. To conclude the proof we need only to adjust the volumes of the sets Fj . For this reason we set F˜j = λj Fj , where λj is chosen so that |F˜j | = |B1 |. Note that λj → 1 since Fj is converging in C 1 to a B1 (x0 ). Observe also that, since P (Fj ) → P (B1 ) and Λ > n, for j large we have P (Fj ) < Λ|Fj |. Therefore for j large we have ˜ − 1| ≤ P (Fj )|λnj − 1| ≤ Λ|λnj − 1||Fj | = Λ |F˜j | − |Fj | . P (Fj ) − P (Fj ) = P (Fj )|λn−1 j From this estimate, recalling (34) we get that C0 (35) P (F˜j ) ≤ P (Fj ) + Λ |F˜j | − |Fj | = P (Fj ) + Λ |Fj | − ωn ≤ P (B1 ) + A(Ej )2 . 2 However, since A(Fj )/A(Ej ) → 1 as j → ∞ we have A(Ej )2 < 2A(F˜j )2 for j large. Therefore, from (35) we obtain P (F˜j ) − P (B1 ) < C0 A(F˜j )2 , which is a contradiction to (30) since, up to translations, the set F˜j have all barycenter at the origin and are converging in C 1 to the unit ball. This contradiction proves the Claim, thus concluding the proof of the quantitative inequality.  5.3. An improved version of the quantitative isoperimetric inequality. Let E be a nearly spherical set and let us look back at the estimate (12) stated in Fuglede’s Theorem 3.6. Observe that in the previous argument we have only used a small part of the information provided by (12), since we have not exploited the presence of the full norm of u in W 1,2 but only its L2 norm. 35

Figure 24. The construction of the oscillation index

The fact that in the quantitative isoperimetric inequality (19) we are throwing away some valuable information encoded in the isoperimetric gap D(E) can be also understood by looking at the two sets E and F in Figure 23. Indeed, E and F have the same measure, the same asymmetry index, but D(E) > 1. Therefore the quantitative isoperimetric inequality (19) gives a sharp information on the set E while gives a completely useless information on the set F . The reason is that while the isoperimetric gap contains also an information on the oscillation of the boundary of the set, the asymmetry index gives only an information on the distance in measure of a set from a ball. This suggests that we should introduce a more precise index which takes into account also the oscillation of the normals to the boundary of the set E. To this aim, given a set of finite perimeter E and a ball Br (y) with the same volume of E, we are going to measure the distance from E to the ball in the following way (see Figure 24). For every point x ∈ ∂ ∗ E we take the projection πy,r (x) of x on the boundary of ∂Br (y) and consider the distance |ν E (x) − ν r,y (πy,r (x))| between the exterior normal to E at the point x and the exterior normal to Br (y) at the projection point πy,r (x). Then, we take the L2 norm of this distance and minimize the resulting norm among all possible balls, thus getting  β(E) := minn y∈R

1 2rn−1

Z

E

|ν (x) − ν ∂∗E

r,y

2

(πy,r (x))| dH

n−1

1/2  (x) .

We shall refer to β(E) as to the oscillation index of the set E. Observe that the Fuglede’s Theorem 3.6 provides indeed an estimate for both the asymmetry and the oscillation index. In fact, if E is a nearly spherical set satisfying (11) with a sufficiently small ε, recall (see Exercise 3.13) that for every point x ∈ ∂ ∗ E the exterior normal to E is given by z(1 + u(z)) − ∇τ u(z) ν E (x) = p , (1 + u(z))2 + |∇τ u(z)|2 36

where z = x/|x| and thus x = z(1 + u(z)). Thus, from (12) we have Z Z  x  n−1 1 x 2 n−1 E 2 E α(E)2 + β(E)2 ≤ |E∆B1 |2 + dH dH = |E∆B | + 1 − ν (x) · ν (x) − 1 2 ∂∗E |x| |x| ∂∗E  Z Z  1 + u(z) 2 n−1 ≤c |u| dH +c 1− p dHn−1 (1 + u)2 + |∇u|2 ∂B1 ∂B1 p Z Z (1 + u)2 + |∇u|2 − (1 + u) 2 n−1 p |u| dH +c =c dHn−1 2 2 (1 + u) + |∇u| ∂B1 ∂B1 Z Z |∇u|2 dHn−1 ≤ cD(E). |u|2 dHn−1 + c ≤c ∂B1

∂B1

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