Elliptic Pdes with Fibered Nonlinearities

Elliptic PDEs with ﬁbered nonlinearities

Ovidiu Savin and Enrico Valdinoci

July 17, 2008

Abstract m n−m 0 00 In R × R , endowed with coordinates x = (x , x ), we consider bounded solutions of the PDE 0 ∆u(x) = f(u(x))χ(x ). We prove a geometric inequality, from which a symmetry result follows.

1 Introduction

In this paper we consider bounded weak solutions u of the equation

∆u(x) = f(u(x))χ(x0), (1)

1 R R Rm Rn where f ∈ C ( ), with f 0 ∈ L∞( ), χ ∈ Llo∞c( ) and x ∈ Ω, for some open set Ω ⊆ . m n m In our notation, x = (x0, x00) = (x1, . . . , xm, xm+1, . . . , xn) ∈ R × R − .

We call the function f(u)χ(x0) a “ﬁbered nonlinearity”. Its main feature is that when χ is constant, the equation in (1) boils down to the usual semilinar equation, while for nonconstant χ the nonlinearity changes only in dependence of a subset of variables.

In particular, χ is constant on the “vertical fibers” {x0 = c}, thence the name of fibered nonlinearity for f(u)χ(x0). The basic model to have in mind is the case in which χ is the characteristic function of a ball. In this sense, (1) may be seen as an interpolation between standard semilinear PDEs and the ones driven by fractional operators (which correspond to PDEs in the halfspace, see [CS07]). In this paper, inspired by similar results in the semilinear case (see [Far02, FSV08]) and in the fractional case (see [CSM05, SV08]), we prove a geometric inequality and a symmetry result. m We introduce some notation. Given a smooth function v, for any fixed x0 ∈ R and c ∈ R, we consider the level set Rn m Lc,x0 (v) := x00 ∈ − s.t. v(x0, x00) = c . m n o Fixed x0 ∈ R , we also define

Rn m Gx0 (v) := x00 ∈ − s.t. ∇x00 v(x0, x00) 6= 0 . (2) n o Notice that Lc,x0 (v) is a smooth (n − m − 1)-dimensional manifolds in the vicinity of the points of Gx0 (v), so we can introduce the principal curvatures on it, denoted by

κ1(x0, x00), . . . , κn m 1(x0, x00). − − We then deﬁne the full curvature K as

n m 1 − − 2 K(x0, x00) := κ (x , x ) . v j 0 00 u j=1 u X t 1 At points x00 ∈ Gx0 (v), we also deﬁne ∇L to be the tangential gradient along Lc,x0 , with c = v(x0, x00), n that is, for any G : R → R smooth in the vicinity of x00, we set

∇x00 v(x0, x00) ∇x00 v(x0, x00) ∇ G(x0, x00) := ∇ G(x0, x00) − ∇ G(x0, x00) · . L x00 x00 |∇ v(x , x )| |∇ v(x , x )| x00 0 00 x00 0 00 Then, the following result holds: Theorem 1. Let u be a bounded weak solution of (1) in an open set Ω ⊆ Rn. Suppose that 2 2 |∇φ(x)| + f 0(u(x))χ(x0) φ (x) dx > 0 (3) ZΩ for any φ ∈ C0∞(Ω). Then,

2 2 2 2 6 2 2 φ |∇x00 u| K + ∇L|∇x00 u| dx00 dx0 |∇x00 u| |∇φ| dx (4) x Rm x G (u) Ω Z 0∈ Z 00∈ x0 Z

for any φ ∈ C0∞(Ω). More precisely,

2 2 2 2 2 6 2 2 φ |∇x00 u| K + ∇L|∇x00 u| dx00 dx0 + φ S dx |∇x00 u| |∇φ| dx, (5) x Rm x G (u) Ω Ω Z 0∈ Z 00∈ x0 Z Z

for any φ ∈ C0∞(Ω), where 2 2 S := ui,j − ∇x0 |∇x00 u| . (6) 16i6m m+1X6j6n

We remark that (5) is indeed more precise (though more complicated) that (4), because S > 0 (see (25) below for further details). The result contained in Theorem 1 may be seen as an extension of similar results presented in [SZ98a, SZ98b] in the classical semilinear framework. We observe that (4) controls the tangential gradients and curvatures of level sets of stable solutions in terms of the gradient of the solution itself, at any n m n m slice of R = R × R − . This means that some geometric quantities of interest are bounded by an appropriate energy term. Another way of looking at (4) is to consider it as a bound of the weighted L2-norm of any test function by a weighted L2-norm of its gradient, at any slice. In this sense, (4) is a kind of weighted Poincar´e inequality. Moreover, one can consider (4) an extension of the stability condition for minimal surfaces, in which the L2-norm of any test function, weighted by the curvature, is controlled by the L2-norm of the gradient (see, for instance, formula (10.20) in [Giu84]). Using Theorem 1, we obtain the following symmetry result for entire solutions (i.e., solutions in the whole space) which have suitably low energy: Theorem 2. Let u be a bounded weak solution of (1) in the whole Rn and suppose that

n ∂nu(x) > 0 for any x ∈ R . (7)

Assume also that there exists C > 0 such that

2 6 2 |∇x00 u| dx CR , (8) ZBR for any R > C.

2 n m 1 ? m Then, there exist ω ∈ S − − and u : R × R → R such that

? u(x0, x00) = u (x0, ω · x00), (9)

m n m for any (x0, x00) ∈ R × R − .

The idea of making use of Poincar´e type inequalities on level sets to deduce suitable symmetries for the solutions has been also used in [Far02, CC06, FSV08]. The motivation for Theorem 2 comes from the many works recently appeared in connection with a conjecture of De Giorgi (see, for instance, [DG79, Mod85, BCN97, GG98, AC00, AAC01, Sav08, FV08]). In particular, Theorem 2 may be thought as an interpolation result between the classical symmetry for semilinear equation and the one for fractional operators (see [CSM05, SV08]), where, after a suitable extension (see [CS07]), the nonlinearity sits as a trace datum for the halfspace. In fact, some involved technical arguments of [SV08] will become here more transparent. In Section 2, we recall some basic regularity theory for solutions of (1). Further motivation on assumptions (3) and (8) will be also given in Section 3. In Sections 4 and 5 we prove Theorem 1 and Theorem 2, respectively.

2 Regularity theory synopsis

We observe that if u is a bounded solution of (1) in a domain Ω, then, by Calder´on-Zygmund theory,

2,p Rn u ∈ Wloc ( ), for any p ∈ (1, +∞). (10) Consequently,

ui ∈ Llo∞c(Ω), for any 1 6 i 6 n. (11) For any ﬁxed ∈ (0, 1) and j, m + 1 6 j 6 n, we deﬁne

u(x + e ) − u(x) v (x) := j .

We point out that v is an incremental quotient in the x00-variables. Therefore, χ(x ) ∆v (x) = 0 f(u(x + e )) − f(u(x)) . (12) j Notice that, given any R > 0, if x ∈ BR, then

χ(x0) f(u(x + ej)) − f(u(x)) 6 kχk kf 0k R kujk , L∞(BR) L∞( ) L∞(BR+1)

so the right hand side of (12) is in Llo∞c(Ω), with norm independent of , thanks to (11). Consequently, Calder´on-Zygmund theory gives that, for any R > 0 and p ∈ (1, +∞),

2,p 6 kvkW (BR) C(R, p),

for some C(R, p) > 0, not depending on . Therefore (see, for instance, Theorem 3 on page 277 of [Eva98]),

2,p uj ∈ Wloc (Ω), for any p ∈ (1, +∞) and any j = m + 1, . . . , n. (13)

3 3 Motivating assumptions (3) and (8)

The goal of this section is to give some geometric condition which is suﬃcient for (3) and (8) to hold.

We begin with a classical observation. For to ∈ R ﬁxed, we set

t F (t) := f(s) ds. (14) Zto Given an open set Ω ⊆ Rn, we also deﬁne

|∇u(x)|2 E (v) := + F (v(x))χ(x0) dx. Ω 2 ZΩ

It is customary to say that u is a local minimizer for EΩ if its energy increases under compact per- turbations, that is, if for any bounded open set U ⊂ Ω we have that EU (u) is well-defined and finite, and EU (u + φ) > EU (u) for any φ ∈ C0∞(U). Lemma 3. If u is a local minimizer in some domain Ω, it satisfies (1) and (3) in Ω.

The proof of Lemma 3 is standard and it follows simply by looking at the ﬁrst and second variation of 7→ EU (u + φ), which is minimal at = 0:

d d2 0 = E (u + φ) 6 E (u + φ) . d U d2 U =0 =0

We now remark that monotonicity in one direction implies stability:

Lemma 4. Let u be a weak solution of (1) in Ω and suppose that ∂nu > 0 in Ω. Then, (3) holds.

The method of the proof of Lemma 4 is also classical (see, for instance [AAC01]): we ﬁx φ ∈ C0∞(Ω), 2 we deﬁne ψ := φ /un, and we use (1), (10), (13) and Cauchy inequality to obtain

2 2 |∇φ| + f 0(u)χ(x0)φ Ω Z 2 2 2 2 2 φ |∇un| φ |∇un| = |∇φ| + − + f 0(u)χ(x0)u ψ u2 u2 n ZΩ n n φ∇φ · ∇u φ2|∇u |2 > n n 2 − 2 + f(u)χ(x0) ψ Ω un un n Z = ∇ψ · ∇un + ∆unψ ZΩ = 0.

This proves Lemma 4. We now give a suﬃcient condition for (8) to hold:

Lemma 5. Let to := −1 in (14). Assume that F (t) > 0 for any t ∈ R, and F (+1) = 0. m m Suppose also that χ ∈ L∞(R ) and χ(x0) > 0 for any x0 ∈ R . Let u be a local minimum in the whole Rn, with |u(x)| 6 1 for any x ∈ Rn.

4 Then, there exists C > 0 such that

2 n 1 |∇u| dx 6 CR − , (15) ZBR for any R > 1. In particular, if also n 6 3, then (8) holds. The proof of Lemma 5 is also standard (see, for instance, [AAC01]): ﬁrst of all, by (11),

Rn 6 k∇ukL∞( ) C0, (16)

for a suitable universal constant C0 > 0.

We then take R > 1, h ∈ C∞(BR), with h = −1 in BR 1, h = 1 on ∂BR and |∇h| 6 10. We set v(x) := min{u(x), v(x)} and we deduce from the minimalit− y of u that

2 2 2 n |∇u| dx 6 EBR (u) 6 EBR (v) = |∇u| + |∇h| + kχkL (R )kF kL ([ 1,1]) dx, ∞ ∞ − BR BR BR 1 Z Z \ − which implies (15) via (16), thus proving Lemma 5. 3 We would like to remark that the ﬁbered Allen-Cahn nonlinearity χ(x0)f(u) = χ(x0)(u − u) may be considered as the typical example satisfying the assumptions of Lemma 5. Below is another criterion for obtaining (8): Lemma 6. Let u be a bounded weak solution of (1) in the whole Rn and suppose that there exist C0 > 0 and σ ∈ [1, 2] such that m σ |χ(x0)| dx0 6 C0R − , (17) x B Z 0∈ R for any R > C0.

Then, there exists C1 > 0 for which

2 n σ |∇u(x)| dx 6 C1R − , (18) x B Z ∈ R for any R > C1. In particular, (8) holds

(P1) if n 6 3 and χ is bounded and supported in {|x10 | 6 C2},

(P2) or if m > 2, n 6 4 and χ is bounded and supported in {|x10 | + |x20 | 6 C2},

for some C2 > 0. The last claim in Lemma 6 plainly follows from (18) (taking σ := 1 in case (P1) holds and σ := 2 in case (P2) holds), so we focus on the proof of (18).

For this, we take R > max{C0, 1} and we deﬁne

Rn M := 1 + kukL∞( ) + sup |f(r)|. r 6 u Rn | | k kL∞( )

We choose τ ∈ C0∞(B2R, [0, 1]), with τ = 1 in BR and |∇τ| 6 10/R. Then, using (1) and Cauchy inequality,

τ 2|∇u|2 = ∇u · ∇(τ 2u) − 2τu∇u · ∇τ ZB2R ZB2R 2 6 f(u)χ(x0)τ u + 2Mτ|∇u| |∇τ| ZB2R

2 1 2 2 2 2 6 M |χ(x0)| + τ |∇u| + 2M |∇τ| . 2 ZB2R

5 Therefore, 1 1 τ 2|∇u|2 6 τ 2|∇u|2 2 2 ZBR ZB2R 2 2 6 2M |χ(x0)| + |∇τ(x)| dx x B Z ∈ 2R 1 6 ¯ 2 n m CM R − |χ(x0)| dx0 + 2 dx x B x B R Z 0∈ 2R Z ∈ 2R 2 n σ n 2 6 C˜M R − + R − , for suitable C˜, C¯ > 0, where (17) has been taken into accoun t. This proves (18) and Lemma 6.

4 Proof of Theorem 1

Given a matrix M = {Mi,j}16i,j6N ∈ Mat(N × N), we now set

2 |M| := Mi,j . (19) 6i,j6N s1 X m Fixed x0 ∈ R , we also deﬁne

Ωx0 := x00 s.t. (x0, x00) ∈ Ω and

Hx0 (u) := x00 ∈ Ωx0 s.t. ∇x00 u(x0, x00) = 0 . (20) Observe, from (2), that n o

Ωx0 = Gx0 (u) ∪ Hx0 (u) , and the union is disjoint. (21)

Now, let φ ∈ C0∞(Ω). From (1), (10), (13) and (19),

2 2 2 f 0(u)χ(x0)|∇x00 u| φ = ∇x00 f(u)χ(x0) · ∇x00 uφ Ω Ω Z Z n 2 2 = ∇x00 (∆u) · ∇x00 uφ = ∆uj ujφ Ω j=m+1 Ω Z X Z n n (22) 2 2 2 2 = − ∇uj · ∇(uj φ ) = − |∇uj| φ + uj∇uj · ∇φ j=m+1 Ω j=m+1 Ω X Z X Z 2 2 2 2 2 1 2 2 = − |D u| φ − u φ − ∇|∇x u| · ∇φ . x00 i,j 2 00 ZΩ 16i6m ZΩ ZΩ m+1X6j6n

We now test (3) against the function |∇x00 u|φ: this function is Lipschitz continuous, due to (13), so we can plug it inside (3), via an easy density argument. We obtain 2 6 2 2 0 ∇ |∇x00 u|φ + f 0(u)χ(x0)|∇x00 u| φ Ω Z (23) 2 2 2 2 1 2 2 2 2 = |∇x u| |∇φ| + ∇|∇x u| φ + ∇|∇x u| · ∇φ + f 0(u)χ(x0)|∇x u| φ . 00 00 2 00 00 ZΩ

6 By comparing (22) and (23), and recalling (6), we see that

2 φ2 |D2 u|2 − ∇ |∇ u| + S 6 |∇ u|2|∇φ|2. (24) x00 x00 x00 x00 ZΩ ZΩ Rm 1,p Fixed x0 ∈ , let now W (x00) := |∇x00 u(x0, x00)|. Since W ∈ Wloc (Ωx0 ) for a.e. x0, because of (13), we deduce from Stampacchia’s Theorem (see, for instance, Theorem 6.19 of [LL97]) that ∇x00 W (x00) = 0 for a.e. x00 ∈ {W = 0}. (j) Analogously, if W (x00) := uj (x0, x00), with m + 1 6 j 6 n, one obtains from Stampacchia’s Theorem (j) (j) that Wi (x00) = 0 for a.e. x00 ∈ {W = 0}. Rm Therefore, recalling (20), for a.e. x0 ∈ and a.e. x00 ∈ Hx0 (u), we have that (j) 6 6 0 = |∇x00 W (x00)| = Wi (x00) for any m + 1 i, j n, or, simply, 2 0 = ∇ |∇ u(x0, x00)| = |D u|. x00 x00 x00 Hence, recalling formula (21) here and formula (2.1) of [SZ98a ],

2 φ2 |D2 u|2 − ∇ |∇ u| x00 x00 x00 Ω Z 2 = φ2 |D2 u|2 − ∇ |∇ u| x00 x00 x00 x Rm x Ω Z 0∈ Z 00∈ x0 2 = φ2 |D2 u|2 − ∇ |∇ u| x00 x00 x00 x Rm x G (u) Z 0∈ Z 00∈ x0 2 2 2 2 = φ |∇x00 u| K + ∇L|∇x00 u| . x Rm x G (u) Z 0∈ Z 00∈ x0 From this and (24), we conclude that (5) holds true. Now we claim that S > 0 a.e.,

and if equality holds at points of {∇x u 6= 0}, 00 (25) then ∇x00 ui is either 0 6 6 or parallel to ∇x00 u, for any 1 i m. Notice that (5) and (25) imply (4), thus the proof of (25) would end the proof of Theorem 1.

In order to prove (25), we focus on points (x0x00) ∈ {|∇x00 u| 6= 0}, since for almost any other point Stampacchia’s Theorem gives that S > 0.

Then, if |∇x00 u| 6= 0, m n 2 1 2 ∇x0 |∇x00 u| = 2 uj ui,j |∇x u| 00 i=1 j=m+1 X X m 1 2 = 2 ∇x00 u · ∇x00 ui |∇x u| 00 i=1 m X 2 6 ∇x00 ui i=1 X 2 = ui,j , 16i6m m+1X6j6n where Cauchy inequality was used. This proves (25) and concludes the proof of Theorem 1.

7 5 Proof of Theorem 2

As usual, we denote by χA the characteristic function of a set A. By Lemma 4, we have that (3) is implied by (7). In particular, in the assumptions of Theorem 2, we Rn m have that Gx0 (u) = − . Following [Far02, FSV08], now we perform the choice of an appropriate test function in Theorem 1. For this, given R > 1, we deﬁne

2 log(R/|x|) φ(x) := χB (x) + χB B (x). √R log R R\ √R We observe that 2 χB B (x) |∇φ(x)| = R\ √R . (26) |x| log R Since φ is Lipschitz continuous, we can use it as a test function in (5), up to a density argument. Therefore, we obtain from (5), (25) and (26) that

2 2 2 |∇x00 u| K + ∇L|∇x00 u| + S x Rm B x B B Z 0∈ ∩ √R/2 Z 00∈ √R/2 Z R (27) 4 |∇ u|2 6 x00 2 2 dx. (log R) B B |x| Z R\ √R We deﬁne 2 E(R) := |∇x00 u| dx. ZBR We now employ Fubini’s Theorem to observe that

2 |∇x00 u| 1 2 dx − 2 E(R) B B 2|x| 2R Z R\ √R |∇ u|2 1 6 x00 2 2 dx − 2 |∇x00 u| dx B B 2|x| 2R B B Z R\ √R Z R\ √R R 2 3 = |∇x00 u| s− ds dx (28) B B x Z R\ √R Z| | R 6 2 3 |∇x00 u| s− dx ds √R B B Z Z s\ √R ! R 3 6 s− E(s) ds. Z√R From (8), (27) and (28),

2 2 2 |∇x00 u| K + ∇L|∇x00 u| + S x Rm B x B B Z 0∈ ∩ √R/2 Z 00∈ √R/2 Z R 8 R C¯ 6 2 3 6 2 R− E(R) + s− E(s) ds , (log R) √R log R Z as long as R > C, for suitable C, C¯ > 0. By taking R arbitrarily large, we thus deduce that

Rm Rn m K = 0 = ∇L|∇x00 u| for any x0 ∈ and x00 ∈ − , (29)

8 and that S = 0. (30)

From (29) and Lemma 2.11 of [FSV08] (applied here at any fixed x0), we infer that, for any fixed m x0 ∈ R , the function x00 7→ u(x0, x00) is a function of one variable, up to rotation. m n m 1 ? m That is, for any fixed x0 ∈ R , there exists ω(x0) ∈ S − − and u : R × R → R such that

? u(x0, x00) = u (x0, ω(x0) · x00), (31)

n for any (x0, x00) ∈ R . We now claim that

∇x00 u(x0, x00) is parallel to ω(x0). (32) n m 1 To check this, we let η(x0) ∈ S − − be orthogonal to ω(x0) and we exploit (31) to get that

? u(x0, x00 + tη(x0)) = u (x0, ω(x0) · x00), and so, by diﬀerentiating with respect to t, we see that

∇x00 u(x0, x00) · η(x0) = 0. This proves (32). In the light of (32), we now write

∇x00 u(x0, x00) = c(x0, x00)ω(x0), (33)

for some c(x0, x00) ∈ R. In fact, (7) and (33) imply that

n c(x0, x00) 6= 0 for all (x0, x00) ∈ R . (34)

Moreover, (13) and (33) give that

2,n+1 Rn 1 Rn the map (x0, x00) 7→ c(x0, x00)ω(x0) belongs to Wloc ( ) ⊂ C ( ). (35) Hence,

c(x0, x00)ω(x0) = ∇x ui(x0, x00), (36) i 0 for any 1 6 i 6 n. Since 2 c (x0, x00) = c(x0x00)ω(x0) · c(x0x00)ω(x0) , we deduce from (33) and (35) that c2 ∈ C1(Rn). As a consequence of this and of (34), we have that

c ∈ C1(Rn). (37)

This, (33) and (34) thus imply that ω ∈ C1(Rm). (38) In particular, 1 ω(x0) · ω(x0) 0 = = = ω (x0) · ω(x0), (39) 2 2 i i i for any 1 6 i 6 m.

From (25), (30), (32) and (36), we have that ∇x00 ui(x0, x00) is parallel to ∇x00 u(x0, x00), that is

(i) c(x0, x00)ω(x0) = k (x0, x00)ω(x0), i 9 (i) for some k (x0, x00) ∈ R. Then, making use of (39) twice, the latter equation gives that

(i) 0 = k (x0, x00)ω(x0) · ωi(x0)

= c(x0, x00)ω(x0) · ωi(x0) i = c(x0, x00)ωi(x0) ·ωi(x0), for any 1 6 i 6 m.

Consequently, from (34), we conclude that ωi(x0) = 0 for any 1 6 i 6 m, and so ω(x0) is constant, say

ω(x0) = ω.

This and (31) give (9), thus ending the proof of Theorem 2.

Acknowledgments

Part of this paper has been written while OS was visiting the Universit`a di Roma Tor Vergata: the hospitality received during this visit is gratefully acknowledged. OS was supported by NSF Grant DMS-07-01037. EV was supported by MIUR Variational Methods and Nonlinear Diﬀerential Equations.

References

[AAC01] Giovanni Alberti, Luigi Ambrosio, and Xavier Cabré. On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property. Acta Appl. Math., 65(1-3):9–33, 2001. Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday. [AC00] Luigi Ambrosio and Xavier Cabré. Entire solutions of semilinear elliptic equations in R3 and a conjecture of De Giorgi. J. Amer. Math. Soc., 13(4):725–739 (electronic), 2000. [BCN97] Henri Berestycki, Luis Caffarelli, and Louis Nirenberg. Further qualitative properties for elliptic equations in unbounded domains. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 25(1- 2):69–94 (1998), 1997. Dedicated to Ennio De Giorgi. [CC06] Xavier Cabré and Antonio Capella. Regularity of radial minimizers and extremal solutions of semilinear elliptic equations. J. Funct. Anal., 238(2):709–733, 2006. [CS07] Luis Caffarelli and Luis Silvestre. An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations, 32(7-9):1245–1260, 2007. [CSM05] Xavier Cabré and Joan Solà-Morales. Layer solutions in a half-space for boundary reactions. Comm. Pure Appl. Math., 58(12):1678–1732, 2005. [DG79] Ennio De Giorgi. Convergence problems for functionals and operators. In Proceedings of the International Meeting on Recent Methods in Nonlinear Analysis (Rome, 1978), pages 131–188, Bologna, 1979. Pitagora. [Eva98] Lawrence C. Evans. Partial differential equations, volume 19 of Graduate Studies in Math- ematics. American Mathematical Society, Providence, RI, 1998.

10 [Far02] Alberto Farina. Propriétés qualitatives de solutions d’équations et systèmes d’équations non-linéaires. Paris VI, 2002. Habilitation a` diriger des recherches. [FSV08] Alberto Farina, Berardino Sciunzi, and Enrico Valdinoci. Bernstein and De Giorgi type problems: new results via a geometric approach. To appear in Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2008. [FV08] Alberto Farina and Enrico Valdinoci. 1D symmetry for solutions of semilinear and quasilinear elliptic equations. Preprint, 2008. http://www.math.utexas.edu/mp arc. [GG98] N. Ghoussoub and C. Gui. On a conjecture of De Giorgi and some related problems. Math. Ann., 311(3):481–491, 1998. [Giu84] Enrico Giusti. Minimal surfaces and functions of bounded variation, volume 80 of Mono- graphs in Mathematics. Birkhäuser Verlag, Basel, 1984. [LL97] Elliott H. Lieb and Michael Loss. Analysis, volume 14 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1997. [Mod85] Luciano Modica. A gradient bound and a Liouville theorem for nonlinear Poisson equations. Comm. Pure Appl. Math., 38(5):679–684, 1985. [Sav08] Ovidiu Savin. Phase transitions: Regularity of flat level sets. Ann. of Math. (to appear), 2008. [SV08] Yannick Sire and Enrico Valdinoci. Fractional Laplacian and boundary reactions phase trnasitions: a geometric inequality and a symmetry result. Preprint, 2008. http://www.math.utexas.edu/mp arc. [SZ98a] Peter Sternberg and Kevin Zumbrun. Connectivity of phase boundaries in strictly convex domains. Arch. Rational Mech. Anal., 141(4):375–400, 1998. [SZ98b] Peter Sternberg and Kevin Zumbrun. A Poincaré inequality with applications to volume- constrained area-minimizing surfaces. J. Reine Angew. Math., 503:63–85, 1998.

Ovidiu Savin Mathematics Department, Columbia University, 2990 Broadway, New York , NY 10027, USA. Email: [email protected]

Enrico Valdinoci Dipartimento di Matematica, Universit`a di Roma Tor Vergata, Via della Ricerca Scientiﬁca 1, 00133 Roma, Italy. Email: [email protected]