EXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES CONVERGING FROM BELOW ON R N. 1. Introduction

Transcription

1 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Abstract. In this paper, we consider the isoperimetric problem in the space R with density. Our result states that, i the density is l.s.c. and converges to a limit a > 0 at ininity, being a ar rom the origin, then isoperimetric sets exist or all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case o our result or radial and increasing densities positively answers a conjecture made in [10]. 1. Introduction In this paper we are interested in the isoperimetric problem with a weight. This means that we are given a positive l.s.c. unction : R R +, usually called density, and we measure volume and perimeter o a generic subset E o R as E := H (E) = (x) dh, P (E) := H ( M E) = E M E (x) dh (x), where the essential boundary o E (which coincides with the usual topological boundary as soon as E is regular) is deined as M E = x R : lim in r 0 H (E B r (x)) ω r < 1 and lim sup r 0 H (E B r (x)) ω r > 0, B r (x) stands or the ball o radius r centered at x, and ω is the euclidean volume o a ball o radius 1. This problem, and many speciic cases, have been extensively studied in the last decades and have many important applications; a short (highly non complete) list o some related papers is [1, 2, 7, 9, 8, 11, 4, 3, 6, 10, 5]. The irst interesting question in this setting is o course the existence o isoperimetric sets, that are sets E with the property that P (E) = J( E ) where, or any V 0, J(V ) := in P (F ) : F = V. Depending on the assumptions on, the answer to this question may be trivial or extremely complicate. Let us start with a very simple, yet undamental, observation. Fix a volume V > 0 and let {E i } be an isoperimetric sequence o volume V : this means that E i = V or every i, and P (E i ) J(V ). Thus, possibly up to a subsequence, the sets E i converge to some set E in the L 1 loc sense. As a consequence, standard lower semi-continuity results in BV ensure that P (E) lim in P (E i ) = J(V ) (at least, or instance, i > 0... ); thereore, i actually E = V, then obviously E is an isoperimetric set. Unortunately, this simple observation is 1

2 2 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI not suicient, in general, to show the existence o isoperimetric sets, because there is no general reason why the volume o E should be exactly V (while it is obviously at most V ). A second remark to be done is the ollowing: i the volume o the whole space R is inite, then in the argument above it becomes obvious that E = V ; basically, the mass cannot vanish to ininity. Hence, in this case isoperimetric sets exist or all volumes. Let us then consider the more general (and interesting) problem when L 1 (R ). this case, by the dierent scaling properties o volume and perimeter, roughly speaking we can say that isoperimetric sets like small density. Let us be a little bit more precise: one can immediately check that, i two dierent balls B 1 and B 2 lie in two regions where the density is constantly d 1 resp. d 2, and i B 1 = B 2, then P (B 1 ) < P (B 2 ) as soon as d 1 < d 2. More in general, all the simplest examples show that isoperimetric sets tend to privilege the zones where the density is lower, and it is very reasonable to expect that this behaviour is quite general. O course, this argument does not predict anything in situations where the density varies quickly (or instance, it would be very convenient or a set to lie where the density is large i at the same time the boundary stays where the density is small!), but nevertheless having this general rule in mind may help a lot. With the aid o the above observation, let us now come back again to the question o the existence o isoperimetric sets. I the density converges to 0 at ininity, one has to expect that isoperimetric sets do not exist (remember that we are assuming R to have ininite volume, otherwise the existence is always true). Indeed, in general a sequence o sets o given volume minimizing the perimeter diverges to ininity, to reach the zones with lowest density, and then actually the inimum o the perimeter or sets o any given volume is generally 0. On the contrary, i the density blows up at ininity, one has to expect isoperimetric sets to exist: indeed, in this case the sequences minimizing the perimeter should remain bounded in order not to go where the density is high, and hence the limit o a minimizing sequence {E i } as above should have volume V, and then it would be an isoperimetric set. A complete answer to this question has been already given in [10]: i the density is also radial, then isoperimetric sets exist or every volume, as expected (Theorem 3.3 in [10]), but i the density is not radial, then the existence might ail (Proposition 5.3 in [10]), contrary to the intuition. Let us then pass to consider the case when the density, at ininity, is neither converging to 0 nor diverging. Again, it is very simple to observe that existence generally ails i the density is decreasing, at least deinitively; analogously, it is easy to build both examples o existence and o non-existence or oscillating densities (that is, densities or which the lim in and the lim sup, at ininity, are dierent). Summarizing, or what concerns the existence problem, the only interesting case let is when the density has a inite limit at ininity, and it is converging to that limit rom below. This leads us to the ollowing deinition. Deinition 1.1. We say that the l.s.c. unction : R R is converging rom below i there exists 0 < a < + such that (x) a when x, and (x) a or x large enough. Basically, the observations above tell that, or unctions which are not converging densities, there is in general no interesting open question about the existence issue. Indeed, as explained In

3 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 3 above, in each o these cases it is already known whether isoperimetric sets exist or all volumes or not. Conversely, or some special cases o densities converging rom below, the existence problem has been already discussed. In particular, combining the results o [10] and [5], the existence o isoperimetric sets ollows or densities which are continuous and converging rom below and which satisy some technical assumptions, or instance it is enough that is superharmonic, or that is radial and or every c > 0 there is some R 1 or which (R) a e cr. Moreover, in [10] it was conjectured that isoperimetric sets exist or all volumes i the density is radial and increasing. In this paper we are able to prove the existence result or any density converging rom below (this is even stronger than the above-mentioned conjecture); as explained above, this result is sharp. Theorem 1.2. Let L 1 loc (R ) be a density converging rom below. Then, isoperimetric sets exist or every volume. 2. General results about isoperimetric sets In this section we present a couple o general acts about existence and boundedness o isoperimetric sets. As already briely described in the Introduction, let us ix some V > 0 and an isoperimetric sequence o volume V, that is, a sequence o sets E j R such that E j = V or any j, and P (E j ) J(V ) or j. As already observed, i (a subsequence o) {E j } converges in L 1 loc to a set E, then by lower semicontinuity P (E) J(V ), and E V ; thus, the set E is automatically isoperimetric o volume V i E = V. However, it is always true that E is isoperimetric or its own volume. We stress that this act is widely known, but we preer to give the proo to keep the presentation sel-contained, and also because we could not ind in the literature any proo which works in such a generality. Ater this lemma, we will show that i there was loss o mass at ininity (that is, i E < V ), then E is necessarily bounded. Lemma 2.1. Assume that L 1 loc (R ) and that is locally bounded rom above ar enough rom the origin. Let {E j } be an isoperimetric sequence o volume V converging in L 1 loc to some set E. Then, E is an isoperimetric set or the volume E. I in addition is converging to some a > 0, then J(V ) = P (E) + (ω a) 1 (V E ). (2.1) Proo. Let us start proving that E is isoperimetric. As we already observed, P (E) J(V ) and E V ; as a consequence, i E = V it is clear that E is isoperimetric, and on the other hand i E = 0 then the empty set E is still clearly isoperimetric or the volume 0. As a consequence, we can assume without loss o generality that 0 < E < V. Suppose now that the claim is alse, and let then F 1 be a set satisying F 1 = E, η := P (E) P (F 1 ) 6 > 0.

4 4 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Select now x R being a point o density 1 in F 1 and a Lebesgue point or with (x) > 0: such a point exists, in particular H -a.e. point o F 1 can be taken. The assumptions on x ensure that, or every radius r small enough, 1 2 ω (x) r B r (x) F 1 B r (x) 2ω (x) r, (2.2) and in turn this implies that there exist arbitrarily small radii r (not necessarily all those small enough) such that H Ä Br (x) ä 2ω (x)r. (2.3) Indeed, i the last inequality were alse or every 0 < r < r, then by integrating we would get that (2.2) is alse. Analogously, let y be a point o density 0 or F 1 which is Lebesgue or with (y) > 0 (the existence o such a point requires that / L 1 (R ), which on the other hand is surely true because E < V ). Since we can ind such a point arbitrarily ar rom the origin (and ar rom x), by assumption it is admissible to assume that M in a small neighborhood o y. As a consequence, there exists some radius ρ > 0 such that, or every 0 < ρ < ρ, B ρ (y) \ F 1 (y) 2 ω ρ, H Ä Bρ (y) ä Mω ρ. (2.4) Let us now ix a constant δ > 0 such that (up to possibly decrease ρ) δ < η, (y) 2 ω ρ > δ, Mω ρ < η. (2.5) We claim the existence o some set F R and o a big constant R > 0 (in particular, much bigger than both x and y ) such that F B R, P (F ) < P (E) 5η, 0 < δ := E F < δ 2, (2.6) writing or brevity B R = B R (0). To show this, it is useul to consider two possible cases. I F 1 is bounded, we deine F = F 1 \ B r (x) or some r very small such that both (2.2) and (2.3) hold true. Then, the inclusion F B R is true or every R big enough, and the two inequalities in (2.6) immediately ollow by (2.2), (2.3) and the deinition o η as soon as r is suiciently small. Instead, i F 1 is not bounded, then we deine F = F 1 B R or a big constant R: o course the inclusion F B R is automatically satisied, and the inequality about δ is also true or every R big enough, say R > R 0. Concerning the inequality on P (F ), i it were alse or every R > R 0, then or every R > R 0 it would be and then by integrating we would get V > F 1 F 1 \ B R0 = H Ä ä F1 B R η, + R 0 H Ä ä F1 B R = +, and the contradiction shows the existence o some suitable R, thus the existence o F satisying (2.6) is proved.

5 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 5 We can now select some R > R such that E \ B R < δ 2, H ( E B R ) > P (E) η. (2.7) Since E j B R (resp., E j B R +1) converges in the L 1 sense to E B R (resp., E B R +1), or every j big enough we have E δ < E j B R E j B R +1 < E + δ, (2.8) H ( E B R ) H ( E j B R ) + η. (2.9) Arguing as above, by (2.8) we have δ > 2δ E j Ä B R +1 \ B R ä = R +1 so we can ind some R j (R, R + 1) such that, also recalling (2.5), Observe that, since E j = V by deinition, (2.8) implies R H (E j B t ) dt, H (E j B Rj ) < δ < η. (2.10) V E δ < E j \ B Rj < V E + δ. As a consequence, calling G j = F Ä E j \ B Rj ä and also recalling (2.6), (2.7), (2.9) and (2.10), we can estimate the volume o G j by and the perimeter o G j by G j = F + E j \ B Rj = E δ + E j \ B Rj (V δ, V ), (2.11) P (G j ) = P (F ) + P (E j \ B Rj ) < P (E) 5η + H ( E j \ B Rj ) + H (E j B Rj ) < H ( E B R ) + H ( E j \ B Rj ) 3η P (E j ) 2η. (2.12) Finally, we deine the competitor E j = G j B ρj (y), where ρ j < ρ is the constant such that E j = V this is possible by (2.11), (2.4), and (2.5). Applying then again (2.4) and (2.5), rom (2.12) we deduce P ( E j ) < P (E j ) η or every j big enough, and this gives the desired contradiction with the act that the sequence E j was isoperimetric. This inally shows that E is an isoperimetric set or the volume E. Let us now pass to the second part o the proo, namely, we assume that is converging to some a > 0 (not necessarily rom below), and we aim to prove (2.1). otice that we can assume without loss o generality that E < V, since otherwise (2.1) is a direct consequence o the act that E is isoperimetric. Arguing as in the irst part o the proo, or every ε > 0 we can ind a very big R such that, calling F = E B R, it is F E ε, P (F ) P (E) + ε.

6 6 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Let then B a ball with volume B = V F : i we take this ball ar enough rom the origin, then B F =, thus G = V being G = F B; moreover, again up to take the ball ar enough, we have a ε a + ε on the whole B. As a consequence, calling r the radius o B, we have rom which we get V E + ε V F = B (a ε)ω r, J(V ) P (G) = P (F ) + P (B) P (E) + ε + (a + ε)ω r P (E) + ε + a + ε (a ε) ω 1 ) (V E + ε, which in turn implies the irst inequality in (2.1) by letting ε 0. To show the other inequality, consider again the isoperimetric sequence {E j }; or any given ε > 0, exactly as in the irst part we can ind an arbitrarily big R so that a ε a + ε out o B R and E B R E ε, P (E \ B R ) ε. For every j 1, then, we can ind some R j (R, R + 1) so that E j B Rj E + ε, H (E j B Rj ) 2ε, P (E) P (E j B Rj ) + 2ε. Since a ε a + ε out o B R, we deduce P (E j \ B Rj ) (a ε)p eucl (E j \ B Rj ) (a ε)ω 1 E j \ B Rj eucl which in turn gives a ε (a + ε) ω 1 E j \ B Rj a ε (a + ε) ω 1 (V E ε P (E j ) = P (E j B Rj ) + P (E j \ B Rj ) 2H (E j B Rj ) P (E) 6ε + a ε (a + ε) ω 1 (V E ε ) ) Since P (E j ) J(V ) or j, by sending ε 0 in the last estimate yields the second inequality in (2.1), thus the proo is concluded. Remark 2.2. Actually, the claim o Lemma 2.1 can be proved even with weaker assumptions; more precisely, one could apply the results o [5] to extend the validity to the more general case when is essentially bounded in the sense o [5]. The second result that we present is a clever observation, which we owe to the courtesy o Frank Morgan, and which shows that whenever a density converges to a limit a > 0 (not necessarily rom below), then i an isoperimetric sequence is losing mass at ininity the remaining limiting set which is isoperimetric thanks to Lemma 2.1 is bounded. Lemma 2.3 (Morgan). Let the density converge to some a > 0, and let the isoperimetric sequence {E j } o volume V converge in L 1 loc to a set E with E < V. Then, E is bounded..,

7 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 7 Proo. Assume that E < V. Then, or every t > 0 deine m(t) = E \ B t = t H (E B σ ) dσ. For every t, we can select a ball B o volume V E + m(t) ar away rom the origin, in order to have no intersection with E B t ; thus, the set (E B t ) B has precisely volume V, hence J(V ) P (E B t ) + P (B). Since the ball B can be taken arbitrarily ar rom the origin, thus in a region where is arbitrarily close to a, exactly as in the second part o the proo o Lemma 2.1 we deduce J(V ) P (E B t ) + (aω ) 1 Ä V E + m(t) ä. Recalling that E < V and comparing the last inequality with (2.1), we obtain P (E) P (E B t ) + Cm(t) or some strictly positive constant C. otice now that P (E) = P (E B t ) + P (E \ B t ) 2H (E B t ) = P (E B t ) + P (E \ B t ) + 2m (t), and in turn by the (Euclidean) isoperimetric inequality i t 1 we have P (E \ B t ) (a ε)p eucl (E \ B t ) (a ε)ω 1 E \ B t eucl Putting everything together, we get Cm(t) 2m (t) + 1 m(t) C 1 a ε (a + ε) ω 1 m(t). or some other constant C 1 > 0. And in turn, i t 1 then m(t) 1, thus the last estimate implies m(t) C 2 Ä m (t) ä. Finally, it is well known that a positive decreasing unction m which satisies the above dierential inequality vanishes in a inite time. Hence, m(t) = 0 or t big enough, and this means precisely that E is bounded. 3. Proo o the main result This section is devoted to show the main result o the paper, namely, Theorem 1.2. Our overall strategy is quite simple, and already essentially contained in [10]. The idea is to take an isoperimetric sequence o volume V, and to consider a limiting set E (up to a subsequence, this is always possible); i E = V, then there is nothing to prove because, as we already saw several times, the set E is already the desired isoperimetric set o volume V. Instead, i E < V, we know by Lemma 2.1 that E is an isoperimetric set or volume E, and by Lemma 2.3 that E is bounded. Moreover, ormula (2.1) says that an isoperimetric set o volume V can be ound as the union o E and a ball at ininity with volume V E. By ball at ininity we mean an hypothetical ball where the density is constantly a: such a ball needs not really to exist, but a sequence o balls o correct volume which escape at ininity will have a perimeter which converges to that o this ball at ininity. In other words, a sequence o sets done by the union

8 8 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI o E and a ball escaping at ininity is isoperimetric thanks to (2.1). Our strategy is then simple: we look or a set B, ar away rom the origin, which is better than a ball at ininity, that is, which has the same volume and less perimeter than it. Since E is bounded (this is a crucial point, rom which the importance o Lemma 2.3) the sets E and B have no intersection, thus the union o E with B is isoperimetric. As one can see, the only thing to do is to ind a set o given volume, arbitrarily ar rom the origin, which is better than a ball at ininity. First o all, let us express in a useul way the act o being better than a ball at ininity, by means o the ollowing deinition. Deinition 3.1. We say that the set E R o inite volume has mean density ρ i P (E) = (ω ρ) 1 E. The meaning o this deinition is evident: ρ is the unique number such that, i we endowe R with the constant density ρ, then balls o volume E have perimeter P (E). The convenience o this notion is also clear: being better than a ball at ininity simply means having mean density less than a. We can then continue our description o the proo o Theorem 1.2: we are let to ind a set o volume V E arbitrarily ar rom the origin and having mean density at most a. Since we want to ind isoperimetric set or any volume V, and we cannot know a priori how much E is, we need to ind sets o mean density less than a o any volume and arbitrarily ar rom the origin. Actually, by a trivial rescaling argument, we can assume that a = 1 and reduce ourselves to search or a set o volume ω. Since is converging to 1 and we must work very ar rom the origin, everything will be very close to the Euclidean case, hence a set o volume ω and mean density less than 1 (or, equivalently, with perimeter less than ω ) must be extremely close to a ball o radius 1. The irst big step in our proo will then be to ind a ball o radius 1 arbitrarily ar rom the origin, and with mean density less than 1. Surprisingly enough, this will by no means conclude the proo, due to a seemingly minor problem: indeed, since converges to 1 rom below, the ball o radius 1 that we have ound does not have exactly volume ω, but only a bit less. And, the urther rom the origin the ball is, the smaller this gap will be, but still positive. otice that at this point we cannot again rely on a rescaling argument: we have already rescaled in order to reduce ourselves to the case o volume ω, but then any other volume will not solve the problem (in principle, it could be that there are sets o mean density less than 1 only or all the rational volumes, and or no irrational one... ). Hence, the second big step in our proo will be to slightly modiy the ball ound in the irst big step, in such a way that the volume increases up to exactly ω, while the mean density remains smaller than 1. At that point, the proo will be concluded. It is to be mentioned that the proo o this second act is more delicate than the one o the irst! Let us now state precisely the claims o the two big steps, and then give the ormal proo o Theorem 1.2 which is more or less exactly what we have just described inormally. Then, we will conclude the paper with two sections, which are devoted to present the proo o the two big claims.

9 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 9 Proposition 3.2. Let be a density converging rom below to 1, and set g = 1. Then, or every ε > 0 there exists a ball B with radius 1 and arbitrarily ar rom the origin such that P g (B) ( ε) B g. Proposition 3.3. Let be a density converging rom below to 1. Then, there exists a set E with volume ω and mean density smaller than 1 arbitrarily ar rom the origin. Proo o Theorem 1.2. Let {E j } be an isoperimetric sequence o volume V, and let E be the L 1 loc limit o a suitable subsequence. I E = V then the proo is already concluded. Otherwise, we know that E is bounded by Lemma 2.3 and that (2.1) holds. Up to a rescaling, we can assume that converges rom below to 1, and that V E = ω. By Proposition 3.3 we can ind a set F not intersecting E with volume ω and mean density less than 1, which means P (F ) ω. The set E F has then volume V, and by (2.1) we obtain P (E F ) J(V ), which means that E F is an isoperimetric set Proo o Proposition 3.2. This section is devoted to the proo o Proposition 3.2. Beore presenting it, it is convenient to show a couple o technical lemmas. Lemma 3.4. Let g : (0, ) [0, ) and α : (, 1) R be L 1 unctions such that 1 lim g(t) = 0, α(t) dt = 0, t σ Then there exists an arbitrarily large R such that 1 α(t)g(t + R) dt 0, with strict inequality unless g(t) = 0 or all t big enough. α(t) dt > 0 σ (, 1). (3.1) Proo. I the claim were alse, then or every choice o R, R with R R + 2 one had 0 > R 1 R = A(R ) + B(R ), α(t)g(t + R) dt dr = R +1 R g(s) s R α(t) dt ds + R +1 R g(s) 1 s R α(t) dt ds where there is no integral over (R +1, R ) because it cancels thanks to (3.1). The conditions on α and g also ensure that A(R ) 0 B(R ) or every R, R. Suppose now that or some arbitrarily large R one has A(R ) > 0; we can then ix R and send R : since g 0, we get B(R ) 0, and then there is some R 1 such that A(R ) + B(R ) > 0, against the above inequality. As a consequence, it must be A(R ) = 0 or every R big enough, and in turn this means that g is deinitively zero, hence any R big enough satisies the claim. Lemma 3.5. Let g : (0, ) [0, ) and β : (, 1) R be L 1 unctions such that g and α(t) = t β(σ) dσ satisy condition (3.1), and α(1) = 0. Then, there exists an arbitrarily large R such that 1 β(t)g(t + R) dt 0, (3.2) with strict inequality unless g(t) = 0 or all t big enough.

10 10 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI Proo. The proo is analogous to the one o Lemma 3.4 above. Take R 1 and assume that the conclusion ails or every R R : then, or every R > R + 2 we have 0 > R 1 R β(t)g(t + R) dt dr = R +1 R s R R +1 1 g(s) β(t) dt ds + g(s) β(t) dt ds. R s R Exactly as beore, since the last term in the right goes to 0 when R, we ind a contradiction as soon as the irst term in the right is strictly positive. In other words, the proo is concluded as soon as we ind some R such that 0 < R +1 R g(s) s R β(t) dt ds = R +1 R g(s)α(s R ) ds = 1 α(t)g(t + R ) dt. And in turn, the existence o such an R is ensured by Lemma 3.4 since α satisies condition (3.1), unless g is deinitively zero. And in this latter case, o course any R big enough would satisy the required condition. We are now in position to prove Proposition 3.2. Proo o Proposition 3.2. For simplicity, we split the proo in two steps: irst we show that one can always reduce himsel to the case o a radial density, and then we prove the claim or this case. Step I. Reduction to radial case. Let us assume that the claim holds or any radial density, and let be not necessarily radial. Deine then the density as the radial average o, namely, (x) = (y) dh (y). (3.3) B x O course, then g = 1 is also the radial average o g. Since the claim holds or the radial density, or any ε > 0 we can ind a ball B satisying P g (B) ( ε) B g. Let us then call B θ, or θ S, the ball having the same distance rom the origin as B, and which is rotated o an angle θ: all the dierent balls B θ are equivalent or the density, but not or the original density. Observe now that by deinition P g (B) = P g (B θ ) dh (θ), S B g = B θ g dh (θ), S and then o course there exists some θ S such that P g (B θ ) ( ε) B θ g. Step II. Proo o the radial case. Thanks to Step I we can assume without loss o generality that is radial. For a ball B R having radius 1 and center at a distance R rom the origin, we can then calculate perimeter and volume by integrating over the radial layers, that is, we have P g (B R ) = 1 ϕ R (t)g(t + R) dt, B R g = 1 ψ R (t)g(t + R) dt, (3.4) where ϕ R (t) and ψ R (t) can be calculated by Fubini Theorem and co-area ormula. Actually, it is not important to write down the exact ormula, while it is immediate to observe that (basically,

11 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 11 since the layers become lat in the limit) the ollowing uniorm limits hold being the limit unctions simply ϕ R (t) ϕ(t) R 1, ψ R (t) ψ(t) R 1, (3.5) ϕ(t) = ( 1)ω (1 t 2 ) 3 2, ψ(t) = ω (1 t 2 ) 2. As a consequence, we can work with the approximated unctions ϕ and ψ in place o ϕ and ψ: more precisely, we call approximated perimeter and volume o B R the unctions P g (B R ) and V g (B) obtained by substituting ϕ and ψ in (3.4) with ϕ and ψ. The claim will be then automatically obtained, thanks to (3.5), i we can ind an arbitrarily large R such that P g (B R ) V g (B R ). We can now deine β : (, 1) R as β(t) = ϕ(t) ψ(t), so that we are reduced to ind an arbitrarily large R such that (3.2) holds. It is elementary to check that the assumptions o Lemma 3.5 are satisied: one can either do the simple calculations, or just observe that α(t) coincides with the perimeter minus times the volume o the portion o the unit ball centered at the origin whose irst coordinate is between and t, so that all the conditions to check become trivial. Thereore, the existence o the searched R directly comes rom Lemma 3.5, and the proo is completed Proo o Proposition 3.3. This last section is entirely devoted to give the proo o Proposition 3.3, which is again divided in some steps. For convenience o the reader, in Steps I and II we start with two particular cases, namely, when is non-decreasing along the hal-lines starting at the origin, and when is radial: even though these two particular cases are not really needed or the proo, the argument is similar to the general one but works more easily, so this helps to understand the general case. Proo o Proposition 3.3. Let us ix ε 1: thanks to Proposition 3.2, there is a ball B = B θ R o radius 1 and centered at the point R θ, with some arbitrarily large R and some θ S, which satisies P g (B) ( ε) B g. Since 1 on B, we have B ω : i B = ω we are already done, because P (B) P eucl (B) = ω, and this automatically implies that the mean density o B is less than 1. Let us then suppose that B < ω, or equivalently that B g > 0, and let us try to enlarge B so to reach volume ω, but still having mean density less than 1. We will do this in some steps. Step I. The case o non-decreasing densities. Let us start with the case when is a non-decreasing density : this means that, or every θ S, the unction t (tθ) is non-decreasing, at least or large t. In this case, let us deine a new set E as ollows. First o all, we decompose B = B l B r, where B l and B r are the let and the right part o the ball B θ R : ormally, a point x B is said to belong to B l or B r i x θ is smaller or bigger than R respectively. Then, or any small δ, we call B l,δ the hal ball centered at (R δ) θ with radius (R δ)/r, and C δ the cylinder o radius 1 and height δ whose axis is the segment connecting (R δ) θ and R θ; inally, we let

12 12 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI E δ = B r B l,δ C δ, see Figure 1, let. Since is converging to 1, and R can be taken arbitrarily big, we have E δ B (1 ε)ω δ ; as a consequence, by continuity we can ix δ such that E = E δ has exactly volume ω, and we have δ (1 + 2ε) B g. (3.6) ω Thanks to the assumption that is non-decreasing, we know that H ( l B l,δ ) H ( l B l ), (3.7) where we call l B l,δ and l B δ the let parts o the boundaries, that is, { l B l = y B l : y θ } { R, l B l,δ = y B l,δ : y θ } R δ. As a consequence, using again that 1 and that R can be taken arbitrarily big, thanks to (3.6) and (3.7) we can evaluate P (E) P (B) + ( 1 + ε)ω δ ω P g (B) + ( 1 + ε)(1 + 2ε) B g ω ( ε) B g + ( 1 + ε)(1 + 2ε) B g < ω. Summarizing, we have built a set E arbitrarily ar rom the origin, with volume exactly ω, and perimeter less than ω, thus mean density less than 1. The proo is then concluded or this case. B l,δ δ/r δ B r E + B + δ E O δ B C δ Figure 1. The sets E o Step I (let) and o Step II (right). The hal-balls B r and B l,δ, as well as the hal-balls B and B + δ, are light shaded; the cylinder C δ, as well as the region E \ (B B δ + ), is dark shaded. Step II. The case o radial densities. Let us now assume that the density is radial. In this case, we cannot use the same argument as in the previous step, because there would be no way to extend the validity o (3.7). evertheless, we can use a similar idea to enlarge the ball B, namely, instead o translating hal o the ball B we rotate it. More ormally, let us take an hyperplane passing through the origin and the center o the ball B θ R, and let us call B ± the two corresponding hal-balls in which B θ R is subdivided. Let us then consider the circle contained in S which contains the direction θ and the direction orthogonal to the hyperplane, and or any small σ > 0 call ρ σ the rotation o an angle σ with respect to this circle. Then, let us call B σ + = ρ σ (B + ) and inally let E δ be the union o B with

13 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 13 all the hal-balls B σ + or 0 < σ < δ, as in Figure 1, right. As in the previous step, since is converging to 1 we can evaluate the dierence o the volumes as E δ B ω (R 1)(1 ε)δ, then we can again select δ such that E = E δ has volume exactly ω and we have δ (1 + 2ε) This time, the radial assumption on gives B g ω (R 1). (3.8) H ( + B δ + ) = H ( + B + ), where we call + B δ + and + B + the upper parts o the boundaries in the obvious sense. And inally, almost exactly as in last step we can evaluate the perimeter o E as P (E) P (B) + ( 1)ω (R + 1) δ ω P g (B) + ( 1)(1 + 2ε) R + 1 R 1 B g ω ( ε) B g + ( 1)(1 + 2ε) R + 1 R 1 B g < ω, where the last inequality again is true i we have chosen ε 1 and then R 1. Thus, the set E has volume ω and mean density less than 1, and the proo is obtained also in this case. Step III. The general case in dimension 2. Let us now treat the case o a general density. For simplicity o notations we assume now to be in the two-dimensional situation = 2, and in the next step we will generalize our argument to any dimension. As in the proo o Proposition 3.2, let us call the radial average o according to (3.3), and g = 1 the radial average o g. Proposition 3.2 provides then us with a ball B R, o radius 1 and distance R 1 rom the origin, such that P g (B R ) ( ε) B R g. (3.9) For any θ S 1, as usual, we call then BR θ the ball o radius 1 centered at Rθ. Let us now argue as in Step II: we call B θ,± R (resp., ± BR θ ) the two hal-balls (resp., hal-circles) made by the points o BR θ (resp., Bθ R ) having direction bigger or smaller than θ; thus, or any small δ > 0, we deine Eδ θ the union o Bθ, R with all the hal-balls Bθ+σ,+ R or 0 < σ < δ. Since the sets Eδ θ are increasing or δ increasing, i R 1 there is a unique δ = δ(θ) such that E θ δ = ω, and exactly as in Step II we have the estimate (3.8) or δ, which or R big enough (since 1 and then g 0) implies δ(θ) (1 + 3ε) Bθ R g ω (R 1). (3.10) Let us then deine the unction τ : S 1 S 1 as τ(θ) = θ + δ(θ), and notice that by construction this is a strictly increasing bijection o S 1 onto itsel, with τ(θ) > θ (i τ(θ) = θ then the ball B θ R has already volume ω, and in this case there is nothing to prove, as already observed). Let us now ix a generic θ S 1, and let η τ(θ) θ: i we call ( A = 0<σ<η Bθ+σ R ) ( \ B θ+η R, B = 0<σ<η Bτ(θ+σ) R ) \ B τ(θ) R,

14 14 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI then, since E θ δ(θ) = ω = E θ+η, δ(θ+η) E θ+η δ(θ+η) = Ä E θ δ(θ) B ä \ A, one has A g = B g. On the other hand, one clearly has B eucl τ(θ + η) τ(θ) =, A eucl η Up to take R big enough, we can assume without loss o generality that 1 ε 1 or points having distance at least R 1 rom the origin, and this yields 1 ε τ(θ + η) τ(η) η 1 1 ε. As an immediate consequence, we get that the unction τ is bi-lipschitz and 1 ε τ (1 ε). Let us now observe that, by construction, all the sets E θ = E θ τ(θ) θ have exactly volume ω : we want then to ind some θ S 1 such that P (E θ) ω, so E θ has mean density less than 1 and we are done. ow, since a simple change o variables gives H Ä g + B θ ä R dθ = H Ä g + B τ(ν) ä S 1 S 1 R τ (ν) dν 1 1 ε H Ä g + B τ(θ) ä S 1 R dθ, we can readily evaluate by (3.9) 0 P g (B R ) ( ε) B R g = P g (BR) θ ( ε) BR θ g dθ S 1 = H Ä g + B θ ä R dθ + H Ä g B θ ä R dθ ( ε) BR θ g dθ S 1 S 1 S 1 1 ε H g Ä + B τ(θ) R BRä θ ( ε) B θ R g dθ, S 1 1 and hence get the existence o some θ S 1 such that H g Thanks to (3.10), we have then Ä Ä P E θä = H + B τ( θ) R ω H g Ä + B τ( θ) R B θ R ä (1 ε)( ε) B θr g. B θ ä ( R + H E θ \ Ä + B τ( θ) R B θ R ä) Ä + B τ( θ) R B θ R ä + ( 1)ω δ( θ)(r + 1) ω (1 ε)( ε) B θ R g + ( 1)(1 + 3ε) B θ R g < ω, where the last inequality holds as soon as ε was chosen small enough at the beginning. The set E θ is then as searched and this step is done. Step IV. The general case. We are now ready to conclude the proo in the general case. We start noticing that in the argument o Step III the assumption = 2 was used only to work with S 1, hence to get the validity o (3.9). More precisely, let us assume that there exists some arbitrarily large R and some circle C S 1 in S such that the estimate P g (BR) θ dh 1 (θ) ( ε) BR θ g dh 1 (θ) (3.11) C C

15 EXISTECE OF ISOPERIMETRIC SETS WITH DESITIES COVERGIG FROM BELOW O R 15 holds true. Then, we can repeat verbatim the proo o Step III, we get the existence o some θ C such that the set E θ R has volume ω and mean density less than 1, and the proo is concluded. Hence, we are let to ind some R and some circle C so that (3.11) holds; notice that, i = 2, then it must be C = S 1 and (3.11) reduces to (3.9), which in turn holds or some arbitrarily large R thanks to Proposition 3.2. Let us then consider the case o dimension = 3. By Proposition 3.2 we can take R 1 such that (3.9) holds true; or any θ S 2, then, we can call C θ the circle in S 2 which is orthogonal to θ, and observe that by homogeneity P g (B R ) = P g (BR) σ dh 1 (σ) dh 2 (θ), C θ S 2 B R g = S 2 C θ B σ R g dh 1 (σ) dh 2 (θ), so thanks to (3.9) we get the existence o a circle C = C θ or which (3.11) holds true: the proo is then concluded also in dimension = 3. otice that the argument above can be rephrased as ollows: i there exists some sphere S S 2 S such that the average estimate (3.11) holds with S in place o C (and in turn in dimension = 3 this reduces to (3.9) and hence holds), then the proo is concluded. As a consequence, the claim ollows also in dimension = 4, arguing exactly as above with the spheres S θ S 2 orthogonal to any θ S 3, and the obvious induction argument gives then the thesis or any dimension. Remark 3.6. otice that, in the proo o Proposition 3.3, we have actually ound a set which has mean density strictly less than 1, unless g 0 on some ball o radius 1. On the other hand, as clearly appears rom the proo o Theorem 1.2, it is impossible to ind such a set i some isoperimetric sequence is losing mass at ininity: indeed, otherwise the argument o Theorem 1.2 would give a set with perimeter strictly less than the inimum. There are then only two possibilities: either there are balls where 1 arbitrarily ar rom the origin, or no isoperimetric sequence can lose mass at ininity. In particular, our proo shows that no isoperimetric sequence can lose mass at ininity i < 1 out o some big ball. Acknowledgment The work o the three authors was supported through the ERC St.G We wish also to thank Michele Marini and Frank Morgan or useul discussions and comments. Reerences [1] F.J. Almgren, Jr., Existence and regularity almost everywhere o solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no [2] F.J. Almgren, Jr., Almgren s big regularity paper. Q-valued unctions minimizing Dirichlet s integral and the regularity o area-minimizing rectiiable currents up to codimension 2. World Scientiic Monograph Series in Mathematics, Vol. 1: xvi+955 pp. (2000). [3] X. Cabré, X. Ros-Oton, J. Serra, Euclidean balls solve some isoperimetric problems with nonradial weights, C. R. Math. Acad. Sci. Paris 350 (2012), no ,

16 16 GUIDO DE PHILIPPIS, GIOVAI FRAZIA, AD ALDO PRATELLI [4] A. Cañete, M. Miranda Jr., D. Vittone, Some isoperimetric problems in planes with density, J. Geom. Anal. 20 (2010), no. 2, [5] E. Cinti, A. Pratelli, The ε ε β property, the boundedness o isoperimetric sets in R n with density, and some applications, to appear in J. Reine Angew. Math. (Crelle), [6] A. Díaz,. Harman, S. Howe, D. Thompson, Isoperimetric problems in sectors with density, Adv. Geom. 12 (2012), no. 4, [7] F. Morgan, Geometric Measure Theory: a Beginner s Guide, Academic Press, 4th edition (2009). [8] F. Morgan, Maniolds with density, otices Amer. Math. Soc. 52 (2005), [9] F. Morgan, D.L. Johnson, Some sharp isoperimetric theorems or Riemannian maniolds, Indiana Univ. Math. J., 49 (2000), no. 3, [10] F. Morgan, A. Pratelli, Existence o isoperimetric regions in R n with density, Ann. Global Anal. Geom. 43 (2013), no. 4, [11] C. Rosales, A. Cañete, V. Bayle, F. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Dierential Equations 31 (2008), no. 1, Institut ür Mathematik Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich (Switzerland) address: guido.dephilippis@math.uzh.ch Department Mathematik, University o Erlangen, Cauerstr. 11, Erlangen (Germany) address: ranzina@math.au.de Department Mathematik, University o Erlangen, Cauerstr. 11, Erlangen (Germany) address: pratelli@math.au.de