Nonlocal Minimal Lawson Cones

arXiv:1303.0593v1 [math.AP] 4 Mar 2013

NONLOCAL MINIMAL LAWSON CONES ´ JUAN DAVILA, MANUEL DEL PINO, AND JUNCHENG WEI Abstract. We prove the existence of the analog of Lawson’s minimal cones for a notion of nonlocal minimal surface introduced by Caffarelli, Roquejoffre and Savin, and establish their stability/instability in low dimensions. In particular we find that there are nonlocal stable minimal cones in dimension 7, in contrast with the case of classical minimal surfaces.

1. Introduction In [4], Caffarelli, Roquejoffre and Savin introduced a nonlocal notion of perimeter of a set E, which generalizes the (N − 1)-dimensional surface area of ∂E. For 0 < s < 1, the s-perimeter of E ⊂ RN is defined (formally) as Z Z dx dy . P ers (E) = |x − y|N +s N E R \E This notion is localized to a bounded open set Ω by setting Z Z Z Z dx dy dx dy − . P ers (E, Ω) = N +s |x − y| |x − y|N +s N N E\Ω R \(E∪Ω) E R \E This quantity makes sense, even if the last two terms above are infinite, by rewriting it in the form Z Z Z Z dx dy dx dy P ers (E, Ω) = + . N +s N +s E∩Ω RN \E |x − y| E\Ω Ω\E |x − y| Let us assume that E is an open set set with ∂E ∩ Ω smooth. The usual notion of perimeter is recovered by the formula lim (1 − s)P ers (E, Ω) = P er(E, Ω) = cN HN −1 (∂E ∩ Ω),

s→1

(1.1)

see [13]. Let us consider a unit normal vector field ν of Σ = ∂E pointing to the exterior of E, and consider functions h ∈ C0∞ (Ω ∩ Σ). For a number t suffiently small, we let Eth be the set whose boundary ∂Eth is parametrized as ∂Eth = {x + th(x)ν(x) / x ∈ ∂E}, with exterior normal vector close to ν. The first variation of the perimeter along these normal perturbations yields Z d HΣs h, P ers (Eth , Ω) =− dt t=0 Σ where

HΣs (p)

:= p.v.

Z

RN

χE (x) − χRN \E (x) dx for p ∈ Σ. |x − p|N +s 1

(1.2)

´ JUAN DAVILA, MANUEL DEL PINO, AND JUNCHENG WEI

2

This integral is well-defined in the principal value sense provided that Σ is regular near p. We say that the set Σ = ∂E is a nonlocal minimal surface in Ω if the surface Σ ∩ Ω is sufficiently regular, and it satisfies the nonlocal minimal surface equation HΣs (p) = 0

for all p ∈ Σ ∩ Ω.

We may naturally call HΣs (p) the nonlocal mean curvature of Σ at p. Let Σ = ∂E be a nonlocal minimal surface. As we will prove in Section 4, the second variation of the s-perimeter in Ω can be computed for functions h smooth and compactly supported in Σ ∩ Ω as Z d2 JΣs [h] h (1.3) P er (E , Ω) = −2 s th dt2 t=0 Σ

where JΣs [h] is the nonlocal Jacobi operator given by JΣs [h](p)

= p.v.

Z

Σ

h(x) − h(p) dx + h(p) |p − x|N +s

Z

Σ

hν(p) − ν(x), ν(p)i dx, |p − x|N +s

p ∈ Σ. (1.4)

In agreement with formula (1.3), we say that an s-minimal surface Σ is stable in Ω if Z JΣs [h] h ≥ 0 for all h ∈ C0∞ (Σ ∩ Ω). − Σ

A basic example of a stable nonlocal minimal surface is a nonlocal area minimizing surface. We say that Σ = ∂E is nonlocal area minimizing in Ω if P ers (E, Ω) ≤ P ers (F, Ω)

(1.5)

for all F such that (E \ F ) ∪ (F \ E) is compactly contained in Ω. In [4], Caffarelli, Roquejoffre and Savin proved that if Ω and E0 ⊂ RN \ Ω are given, and sufficiently regular, then there exists a set E with E ∩ (RN \ Ω) = E0 which satisfies (1.5). They proved that Σ = ∂E is a surface of class C 1,α outside a closed set of Hausdorff dimension N − 2. In this paper we will focus our attention on nonlocal minimal cones. By a (solid) cone in RN , we mean a set of the form E = {tx / t > 0, x ∈ O}

where O is a regular open subset of the sphere S N −1 . The cone (mantus) Σ = ∂E is an (N − 1)-dimensional surface which is regular, except at the origin. Existence or non-existence of area minimizing cones for a given dimension is a crucial element in the classical regularity theory of minimal surfaces. Simons [15] proved that no stable minimal cone exists in dimension N ≤ 7, except for hyperplanes. This result is a main ingredient in regularity theory: it implies that area minimizing surfaces must be smooth outside a closed set of Hausdorff dimension N − 8. Savin and Valdinoci [13], by proving the nonexistence of a nonlocal minimizing cone in R2 , established the regularity of any nonlocal minimizing surface outside a set of Hausdorff dimension N − 3, thus improving the original result in [4]. In [5], Caffarelli and Valdinoci proved that regularity of non-local minimizers holds up to a (N − 8)-dimensional set, provided that s is sufficiently close to 1.

NONLOCAL MINIMAL LAWSON CONES

3

The purpose of this paper is to analyze a specific class of nonlocal minimal cones. Let n, m ≥ 1, n + m = N and α > 0. Let us call Cα = {x = (y, z) ∈ Rm × Rn / |z| = α|y| }.

(1.6)

It is a well-known fact that Cα is a minimal surface in RN \ {0} (its mean curvature equals zero) if and only r n−1 n ≥ 2, m ≥ 2, α = . m−1

n We call this minimal Lawson cone Cm ([11]). As for the stability-minimizing character of these cones, the result of Simons [15] tells us that they are all unstable for m + n ≤ 7. Simons also proved that the cone C44 is stable and conjectured that it was minimizing. Bombieri, De Giorgi and Giusti in [3] found a family of disjoint minimal surfaces asymptotic to the cone, foliating R4 × R4 . This implies γ = C44 n is area minimizing. For N > 8 the cones Cm are all area minimizing. For N = 8 they are area minimizing if and only if |m − n| ≤ 2. These facts were established by Lawson [11] and Simoes [14], see also [12, 6, 1, 8]. n For the non-local scenario we find the existence of analogs of the cones Cm .

Theorem 1. For any given m ≥ 1, n ≥ 1, 0 < s < 1, there is a unique α = α(s, m, n) > 0 such that Cα = {x = (y, z) ∈ Rm × Rn / |z| = α|y| } is a nonlocal n minimal cone. We call this cone Cm (s). 1 The above result includes the existence of a minimal cone Cm (s), m ≥ 1. Such n an object does not exist in the classical setting for Cm is defined only if n, m ≥ 2. n We have found a (computable) criterion to decide whether or not Cm (s) is stable. As a consequence we find the following result for s close to 0 which shows a sharp contrast with the classical case.

Theorem 2. There is a s0 > 0 such that for each s ∈ (0, s0 ), all minimal cones n (s) are unstable if N = m + n ≤ 6 and stable if N = 7. Cm n is unstable for N = 7. It is natural to We recall that in the classical case Cm conjecture that the above cones for N = 7 are minimizers of perimeter. Being that the case, the best regularity possible for small s would be up to an (N − 7)dimensional set.

As far as we know, at this moment, there are no examples of regular nontrivial nonlocal minimal surfaces ([16]). Formula (1.1) suggests that for s close to 1 there may be nontrivial nonlocal minimal surfaces close to the classical ones. In a forthcoming paper [7] we prove that this is indeed the case. We construct nonlocal catenoids as well as nonlocal Costa surfaces for s close to 1 by interpolating the classical minimal surfaces in compact regions with the nonlocal Lawson’s cones 1 Cm far away. Thus these nonlocal catenoids can be considered as foliations of the 1 nonlocal Lawson’s cones Cm . A natural question, as in the classical minimal cones n case ([8]), is the existence of foliations for general nonlocal Lawson’s cones Cm . In section 2 we prove theorem 1 and in section 3 we show that also for s = 0 there is a unique minimal cone. In section 4 we obtain formula (1.4) for the nonlocal Jacobi operator and section 5 is devoted to the proof of theorem 2.


4

2. Existence and uniqueness Let us write Eα = {x = (y, z) : y ∈ Rm , z ∈ Rn , |z| > α|y| },

(2.1)

so that Cα = ∂Eα is the cone defined in (1.6). Proof of theorem 1. Existence. We fix N , m, n with N = m + n, n ≤ m and also fix 0 < s < 1. If m = n then C1 is a minimal cone, since (1.2) is satisfied by symmetry. So we concentrate next on the case n < m. Before proceeding we remark that for a cone Cα the quantity appearing in (1.2) has a fixed sign for all p ∈ Cα , p 6= 0, since by rotation we can always assume that p = rpα for some r > 0 where pα = √

1 (m) (n) (e1 , αe1 ) 1 + α2

with (m)

e1

= (1, 0, . . . , 0) ∈ Rm

(2.2)

(n)

and similarly for e1 . Then we observe that Z Z χEα (x) − χEαc (x) χEα (x) − χEαc (x) 1 dx = s p.v. dx. p.v. N +s |x − rp | r |x − pα |N +s α RN RN Let us define H(α) = p.v.

Z

RN

χEα (x) − χEαc (x) dx |x − pα |N +s

(2.3)

and note that it is a continuous function of α ∈ (0, ∞). Claim 1. We have H(1) ≤ 0.

(2.4)

Indeed, write y ∈ Rm as y = (y1 , y2 ) with y1 ∈ Rn and y2 ∈ Rm−n . Abbreviating (n) e1 = e1 = (1, 0, . . . , 0) ∈ Rn we rewrite Z χE1 (x) − χE1c (x) dx H(1) = lim δ→0 RN \B(p ,δ) |x − p1 |N +s 1 Z 1 = lim N +s δ→0 A (|y − √1 e |2 + |y |2 + |z − √1 e |2 ) 2 δ 2 1 2 1 2 1 Z 1 − lim N +s , δ→0 B (|y − √1 e |2 + |y |2 + |z − √1 e |2 ) 2 δ 2 1 2 1 2 1 where 1 Aδ = {|z|2 > |y1 |2 + |y2 |2 , |y1 − √ e1 |2 + |y2 |2 + |z − 2 1 Bδ = {|z|2 < |y1 |2 + |y2 |2 , |y1 − √ e1 |2 + |y2 |2 + |z − 2

1 √ e 1 |2 > δ 2 } 2 1 √ e1 |2 > δ 2 }. 2


But the first integral can be rewritten as Z Aδ

=

Z

√1 e1 |2 2

(|y1 −

˜δ A

1

+ |y2 |2 + |z −

√1 e1 |2 2

(|y1 −

5

1

N +s √1 e1 |2 ) 2 2

+ |y2 |2 + |z −

N +s √1 e1 |2 ) 2 2

where 1 1 A˜δ = {|y1 |2 > |z|2 + |y2 |2 , |y1 − √ e1 |2 + |y2 |2 + |z − √ e1 |2 > δ 2 } 2 2 (we just have exchanged y1 by z and noted that the integrand is symmetric in these variables). But A˜δ ⊂ Bδ and so Z χE1 (x) − χE1c (x) dx |x − p1 |N +s RN \B(p1 ,δ) Z 1 =− N +s ≤ 0. 1 ˜δ (|y1 − √ e1 |2 + |y2 |2 + |z − √1 e1 |2 ) 2 Bδ \A 2 2 This shows the validity of (2.4). Claim 2. We have H(α) → +∞ as α → 0.

(2.5)

Let 0 < δ < 1/2 be fixed and write H(α) = Iα + Jα where Iα =

Z

RN \B(pα ,δ)

Jα = p.v.

Z

B(pα ,δ)

χEα (x) − χEαc (x) dx |x − pα |N +s

χEα (x) − χEαc (x) dx. |x − pα |N +s

With δ fixed lim Iα =

α→0

Z

RN \B(pα ,δ)

1 dx > 0. |x − p0 |N +s

(2.6)

For Jα we make a change of variables x = α˜ x + pα and obtain Z Z x) − χFαc (˜ x) χEα (x) − χEαc (x) χFα (˜ 1 dx = p.v. d˜ x Jα = p.v. N +s s N +s |x − pα | α |˜ x| B(pα ,δ) B(0,δ/α) (2.7) where Fα =

1 α (Eα

p.v.

Z

− pα ). But

B(0,δ/α)

x) − χFαc (˜ x) χFα (˜ d˜ x → p.v N +s |˜ x|

Z

RN

χF0 (x) − χF0c (x) dx |x|N +s

6

´ JUAN DAVILA, MANUEL DEL PINO, AND JUNCHENG WEI (n)

as α → 0 where F0 = {x = (y, z) : y ∈ Rm , z ∈ Rn , |z + e1 | > 1}. But writing z = (z1 , . . . , zn ) we see that Z Z χ[z1 >0 or z1 0, associated to solid cones Eα1 and Eα2 . We claim that there is a rotation R so that R(Eα1 ) ⊂ Eα2 (strictly) and that Z Z χR(Eα1 ) (x) − χR(Eα1 )c (x) dx. H(α1 ) = p.v. |x − pα2 |N +s RN RN

Note that the denominator in the integrand is the same that appears in (2.3) for α2 and then Z Z χR(Eα1 ) (x) − χR(Eα1 )c (x) dx H(α1 ) = p.v. |x − pα2 |N +s RN RN Z Z χ c (x) Eα2 (x) − χEα 2 < p.v. dx = H(α2 ). (2.8) |x − pα2 |N +s RN RN

This shows that H(α) is decreasing in α and hence the uniqueness. To construct the rotation let us write as before x = (y, z) ∈ RN , with y ∈ Rm , z ∈ Rn , and y = (y1 , y2 ) with y1 ∈ Rn , y2 ∈ Rm−n (we assume alway n ≤ m). Let us write the vector (y1 , z) in spherical coordinates of R2n as follows   cos(ϕ1 )   sin(ϕ1 ) cos(ϕ2 )     sin(ϕ ) sin(ϕ ) cos(ϕ ) 1 2 3 y1 = ρ     ..   . sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) . . . sin(ϕn−1 ) cos(ϕn )   sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) . . . sin(ϕn ) cos(ϕn+1 )   ..   . z = ρ  sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) . . . sin(ϕ2n−2 ) cos(ϕ2n−1 ) sin(ϕ1 ) sin(ϕ2 ) sin(ϕ3 ) . . . sin(ϕ2n−2 ) sin(ϕ2n−1 ) where ρ > 0, ϕ2n−1 ∈ [0, 2π), ϕj ∈ [0, π] for j = 1, . . . , 2n − 2. Then |z|2 = ρ2 sin(ϕ1 )2 sin(ϕ2 )2 . . . sin(ϕn )2 ,

|y1 |2 + |z|2 = ρ2 .

The equation for the solid cone Eαi , namely |z| > αi |y|, can be rewritten as ρ2 sin(ϕ1 )2 sin(ϕ2 )2 . . . sin(ϕn )2 > α2i (|y1 |2 + |y2 |2 ).

Adding α2i |z|2 to both sides this is equivalent to

sin(ϕ1 )2 sin(ϕ2 )2 . . . sin(ϕn )2 > sin(βi )2 (1 +

|y2 |2 ) ρ2


7

where βi = arctan(αi ). We let θ = β1 − β2 ∈ (0, π/2), and define the rotated cone Rθ (Eα1 ) by the equation sin(ϕ1 + θ)2 sin(ϕ2 )2 . . . sin(ϕn )2 > sin(β1 )2 (1 +

|y2 |2 ). ρ2

We want to show that Rθ (Eα1 ) ⊂ Eα2 . To do so, it suffices to prove that for any given t ≥ 1, if ϕ satisfies the inequality | sin(ϕ + θ)| > sin(β1 )t then it also satisfies | sin(ϕ)| > sin(β2 )t. This in turn can be proved from the inequality arccos(sin(β1 )t) + θ < arccos(sin(β2 )t) 1 . For t = 1 we have equality 1 < t ≤ sin(β 1) 1 1 < t ≤ sin(β1 ) can be checked by computing

by definition of θ. The inequality for for a derivative with respect to t. The strict inequality in (2.8) is because R(Eα1 ) ⊂ Eα2 strictly. 3. Minimal cones for s = 0 In this section we derive the limiting value α0 = lims→0 αs where αs is such that Cαs is an s-minimal cone. Proposition 3.1. Assume that n ≤ m in (2.1), N = m + n. The number α0 is the unique solution to Z α Z ∞ tn−1 tn−1 dt = 0. N dt − 2 N (1 + t2 ) 2 0 (1 + t ) 2 α Proof. We write x = (y, z) ∈ RN with y ∈ Rm , z ∈ Rn . Let us assume in the rest of the proof that n ≥ 2. The case n = 1 is similar. We evaluate the integral in (1.2) (m) (n) for the point p = (e1 , αe1 ) using spherical coordinates for y = rω1 and z = ρω2 where r, ρ > 0 and   cos(θ1 )   sin(θ1 ) cos(θ2 )     .. ω1 =  (3.1)  .   sin(θ1 ) sin(θ2 ) . . . sin(θm−2 ) cos(θm−1 ) sin(θ1 ) sin(θ2 ) . . . sin(θm−2 ) sin(θm−1 ) 

cos(ϕ1 ) sin(ϕ1 ) cos(ϕ2 ) .. .



      ω2 =  ,   sin(ϕ1 ) sin(ϕ2 ) . . . sin(ϕn−2 ) cos(ϕn−1 ) sin(ϕ1 ) sin(ϕ2 ) . . . sin(ϕn−2 ) sin(ϕn−1 )

(3.2)

where θj ∈ [0, π] for j = 1, . . . , m − 2, θm−1 ∈ [0, 2π], ϕj ∈ [0, π] for j = 1, . . . , n − 2, ϕn−1 ∈ [0, 2π]. Then (m)

(n)

|(y, z) − (e1 , αe1 )|2 = r2 + 1 − 2r cos(θ1 ) + ρ2 + α2 − 2ρα cos(ϕ1 ).

Assuming that α = αs > 0 is such that Cαs is an s-minimal cone, (1.2) yields the following equation for α Z ∞ rm−1 (Aα,s (r) − Bα,s (r))dr = 0 (3.3) p.v. 0


8

where Aα,s (r) = Bα,s (r) =

∞

Z

π

Z

π

Z

rα 0 0 Z rα Z π Z π 0

0

0

ρn−1 sin(θ1 )m−2 sin(ϕ1 )n−2 (r2 + 1 − 2r cos(θ1 ) + ρ2 + α2 − 2ρα cos(ϕ1 )) ρn−1 sin(θ1 )m−2 sin(ϕ1 )n−2

N +s 2

(r2 + 1 − 2r cos(θ1 ) + ρ2 + α2 − 2ρα cos(ϕ1 ))

N +s 2

dθ1 dϕ1 dρ dθ1 dϕ1 dρ,

which are well defined for r 6= 1. Setting ρ = rt we get Aα,s (r) =r

−m−s

Z

∞ α

= cm,n r−m−s

Z

π

0

Z

∞

α

Z

0

π

tn−1 sin(ϕ1 )m−2 sin(θ1 )n−2 1 r2

(1 + tn−1

(1 +

t2 )

−

N +s 2

2 r

α2 r2

cos(θ1 ) + t2 +

− 2r tα cos(ϕ1 ))

N +s 2

dθ1 dϕ1 dt

dt + O(r−m−s−1 )

as r → ∞ and this is uniform in s for s > 0 small. Here cm,n > 0 is some constant. Similarly Z α tn−1 −m−s−1 Bα,s (r) = cm,n r−m−s ) N +s dt + O(r 0 (1 + t2 ) 2 Then (3.3) takes the form Z Z ∞ Z 2 . . . dr = O(1) + Cs (α) . . . dr + 0= 0

2

where

Cs (α) =

Z

∞

tn−1

r−1−s dr = O(1) +

2

Z

α

2−s Cs (α) s

tn−1

N +s dt (1 + t2 ) 2 0 (1 + t2 ) 2 and O(1) is uniform as s → 0, because 0 < αs ≤ 1 by theorem 1, and the only singularity in (3.3) occurs at r = 1. This implies that α0 = lims→0 αs has to satisfy C0 (α0 ) = 0.

α

N +s dt −

∞

4. The Jacobi operator In this section we prove formula (1.3) and derive the formula for the nonlocal Jacobi operator (1.4). Let E ⊂ RN be an open set with smooth boundary and Ω be a bounded open set. Let ν be the unit normal vector field of Σ = ∂E pointing to the exterior of E. Given h ∈ C0∞ (Ω ∩ Σ) and t small, let Eth be the set whose boundary ∂Eth is parametrized as ∂Eth = {x + th(x)ν(x) / x ∈ ∂E}, with exterior normal vector close to ν. Proposition 4.1. For h ∈ C0∞ (Ω ∩ Σ) Z Z d2 s P ers (Eth , Ω) = −2 JΣ [h] h − h2 HHΣs , dt2 t=0 Σ Σ

(4.1)

where JΣs is the nonlocal Jacobi operator defined in (1.4), H is the classical mean curvature of Σ and HΣs is the nonlocal mean curvature defined in (1.2). In case that Σ is a nonlocal minimal surface in Ω we obtain formula (1.3). Another related formula is the following.


9

Proposition 4.2. Let Σth = ∂Eth . For p ∈ Σ fixed let pt = p + th(p)ν(p) ∈ Σth . Then for h ∈ C ∞ (Σ) ∩ L∞ (Σ) d s HΣth (pt ) = 2JΣs [h](p). (4.2) dt t=0 A consequence of proposition 4.2 is that entire nonlocal minimal graphs are stable. Corollary 4.1. Suppose that Σ = ∂E with E = {(x′ , F (x′ )) ∈ RN : x′ ∈ RN −1 } is a nonlocal minimal surface. Then Z − JΣs [h] h ≥ 0

for all

Σ

h ∈ C0∞ (Σ).

(4.3)

Proof of proposition 4.1. Let Kδ (z) =

1

ηδ (z) |z|N +s

where ηδ (x) = η(x/δ) (δ > 0) and η ∈ C ∞ (RN ) is a radially symmetric cut-off function with η(x) = 1 for |x| ≥ 2, η(x) = 0 for |x| ≤ 1. Consider Z Z Z Z Kδ (x − y)dydx. Kδ (x − y) dy dx + P ers,δ (Eth , Ω) = Eth ∩Ω

RN \Eth

Eth \Ω

Ω\Eth

(4.4)

2

d We will show that dt 2 P ers,δ (Eth , Ω) approaches a certain limit D2 (t) as δ → 0, uniformly for t in a neighborhood of 0 and that Z Z s h2 HHΣs . JΣ [h] h − D2 (0) = −2 Σ

Σ

First we need some extensions of ν and h to RN . To define them, let K ⊂ Σ be the support of h and U0 be an open bounded neighborhood of K such that for any x ∈ U0 , the closest point x ˆ ∈ Σ to x is unique and defines a smooth function of x. We also take U0 smaller if necessary as to have U 0 ⊂ Ω. Let ν˜ : RN → RN be a globally defined smooth unit vector field such that ν˜(x) = ν(ˆ x) for x ∈ U0 . ˜ : RN → R such that it is smooth with compact support We also extend h to h ˜ contained in Ω and h(x) = h(ˆ x) for x ∈ U0 . From now one we omit the tildes (˜) in the definitions of the extensions of ν and h. For t small x¯ 7→ x¯ + th(¯ x)ν(¯ x) is a global diffeomorphism in RN . Let us write u(¯ x) = h(¯ x)ν(¯ x) ν = (ν 1 , . . . , ν N ),

for x ¯ ∈ RN , u = (u1 , . . . , uN )

and let Jt (¯ x) = Jid+tu (¯ x) be the Jacobian determinant of id + tu. We change variables x=x ¯ + tu(¯ x),

y = y¯ + tu(¯ y),


10

in (4.4) P ers,δ (Eth , Ω) =

Z

E∩φt (Ω)

+

Z

Z

RN \E

E\φt (Ω)

Z

Kδ (x − y)Jt (¯ x)Jt (¯ y )d¯ y d¯ x,

φt (Ω)\E

Kδ (x − y)Jt (¯ y )d¯ y d¯ x,

where φt is the inverse of the map x ¯ 7→ x¯ + tu(¯ x). Differentiating with respect to t: Z Z h d ∇Kδ (x − y)(u(¯ x) − u(¯ y))Jt (¯ x)Jt (¯ y) P ers,δ (Eth , Ω) = dt E∩φt (Ω) RN \E i + Kδ (x − y)(Jt′ (¯ x)Jt (¯ y ) + Jt (¯ x)Jt′ (¯ y )) d¯ y d¯ x Z Z h ∇Kδ (x − y)(u(¯ x) − u(¯ y ))Jt (¯ x)Jt (¯ y) + E\φt (Ω)

φt (Ω)\E

i + Kδ (x − y)(Jt′ (¯ x)Jt (¯ y ) + Jt (¯ x)Jt′ (¯ y )) d¯ y d¯ x,

where

d Jt (¯ x). dt Note that there are no integrals on ∂φt (Ω) for t small because u vanishes in a neighborhood of ∂Ω. d P ers,δ (Eth , Ω) have compact support contained in Since the integrands in dt φt (Ω) (t small), we can write Z Z h d ∇Kδ (x − y)(u(¯ x) − u(¯ y))Jt (¯ x)Jt (¯ y) P ers,δ (Eth , Ω) = dt E RN \E i + Kδ (x − y)(Jt′ (¯ x)Jt (¯ y ) + Jt (¯ x)Jt′ (¯ y )) d¯ y d¯ x. Jt′ (¯ x) =

Differentiating once more

d2 P ers,δ (Eth , Ω) = A(δ, t) + B(δ, t) + C(δ, t) dt2 where A(δ, t) =

Z Z E

RN \E

Z Z

B(δ, t) = 2 E Z Z C(δ, t) = E

D2 Kδ (x − y)(u(¯ x) − u(¯ y ))(u(¯ x) − u(¯ y ))Jt (¯ x)Jt (¯ y )d¯ y d¯ x

RN \E

RN \E

∇Kδ (x − y)(u(¯ x) − u(¯ y ))(Jt′ (¯ x)Jt (¯ y ) + Jt (¯ x)Jt′ (¯ y ))d¯ y d¯ x

x)Jt′ (¯ y ) + Jt (¯ x)Jt′′ (¯ y ))d¯ y d¯ x. Kδ (x − y)(Jt′′ (¯ x)Jt (¯ y ) + 2Jt′ (¯

We claim that A(δ, t), B(δ, t) and C(δ, t) converge as δ → 0 for uniformly for t near 0, to limit expressions A(0, t), B(0, t) and C(0, t), which are the same as above replacing δ by 0, and that the integrals appearing in A(0, t), B(0, t) and C(0, t) are well defined. Indeed, we can estimate Z Z 1 dy dx, |A(δ, t) − A(0, t)| ≤ C |x − y|N +s c x∈E∩K0 y∈E ,|x−y|≤2δ


11

where K0 is a fixed bounded set. For x ∈ E ∩ K0 we see that Z C 1 dy ≤ , N +s dist(x, E c )s y∈E c ,|x−y|≤2δ |x − y| and therefore |A(δ, t) − A(0, t)| ≤ C ≤ C

Z

x∈E∩K0 , dist(x,E c )≤2δ

1 dx ≤ Cδ 1−s . dist(x, E c )s

The differences B(δ, t) − B(0, t), C(δ, t) − C(0, t) can be estimated similarly. This shows that d2 d2 P er (E , Ω) P er (E , Ω) = lim = lim A(δ, 0) + B(δ, 0) + C(δ, 0). s th s,δ th δ→0 dt2 δ→0 dt2 t=0 t=0 In what follows we will evaluate A(δ, 0) + B(δ, 0) + C(δ, 0). At t = 0 we have Z Z A(δ, 0) = Dxi xj Kδ (x − y)(ui (x) − ui (y))(uj (x) − uj (y)) dy dx E

RN \E

= A11 + A12 + A21 + A22

where A11 =

Z Z E

A12 = −

RN \E

Dxi xj Kδ (x − y)ui (x)uj (x) dy dx

Z Z

RN \E

E

Z Z

A21 = − E Z Z A22 =

RN \E

E

Let us also write Z Z B(δ, 0) = 2 E

RN \E

RN \E

Dxi xj Kδ (x − y)ui (x)uj (y) dy dx Dxi xj Kδ (x − y)ui (y)uj (x) dy dx

Dxi xj Kδ (x − y)ui (y)uj (y) dy dx.

Dxj Kδ (x − y)(uj (x) − uj (y))(div(u)(x) + div(u)(y)) dy dx

= B11 + B12 + B21 + B22 ,

where B11 = 2

Z Z E

B12 = 2

RN \E

Z Z E

RN \E

Z Z

B21 = −2 E Z Z B22 = 2 E

Dxj Kδ (x − y)uj (x)div(u)(x) dy dx Dxj Kδ (x − y)uj (x)div(u)(y) dy dx

RN \E

RN \E

Dxj Kδ (x − y)uj (y)div(u)(x) dy dx

Dyj Kδ (x − y)uj (y)div(u)(y) dy dx,

and C(δ, 0) = C1 + C2 + C3 ,


12

where C1 =

Z Z

RN \E

E

C2 =

Z Z

RN \E

E

Z Z

C3 = 2

E

h i Kδ (x − y) div(u)(x)2 − tr(Du(x)2 ) dy dx h i Kδ (x − y) div(u)(y)2 − tr(Du(y)2 ) dy dx

RN \E

Kδ (x − y)div(u)(x)div(u)(y) dy dx.

We compute Z Z i h A11 = Dxi Dxj Kδ (x − y)ui (x)uj (x) dy dx E RN \E Z Z h i Dxj Kδ (x − y)Dxi ui (x)uj (x) dy dx − E RN \E Z Z Dxj Kδ (x − y)ui (x)uj (x)ν i (x) dy dx = N ∂E R \E Z Z i h − Dxj Kδ (x − y) Dxi ui (x)uj (x) + ui (x)Dxi uj (x) dy dx. E

RN \E

Therefore

A11 + B11 =

Z

∂E

+

Z

RN \E

Dxj Kδ (x − y)ui (x)uj (x)ν i (x) dy dx

Z Z E

RN \E

i h Dxj Kδ (x − y) Dxi ui (x)uj (x) − ui (x)Dxi uj (x) dy dx.

We express the first term as Z Z Dxj Kδ (x − y)ui (x)uj (x)ν i (x) dy dx N ∂E R \E Z Z Dyj Kδ (x − y)ui (x)uj (x)ν i (x) dy dx =− N ∂E R \E Z Z Kδ (x − y)ui (x)uj (x)ν i (x)ν j (y) dy dx = ∂E ∂E Z Z Kδ (x − y)h(x)2 ν(x)ν(y) dy dx. = ∂E

∂E

For the second term of A11 + B11 let us write Z Z Dxj Kδ (x − y)Dxi ui (x)uj (x) dy dx E RN \E Z Z i h Dxj Kδ (x − y)Dxi ui (x)uj (x) dy dx = E RN \E Z Z i h − Kδ (x − y)Dxj Dxi ui (x)uj (x) dy dx E RN \E Z Z Kδ (x − y)Dxi ui (x)uj (x)ν j (x) dy dx = ∂E RN \E Z Z i h Kδ (x − y) Dxj xi ui (x)uj (x) + div(u)(x)2 dy dx. − E

RN \E


13

The third term of A11 + B11 is −

Z Z E

RN \E

Dxj Kδ (x − y)ui (x)Dxi uj (x) dy dx Z Z i h Dxj Kδ (x − y)ui (x)Dxi uj (x) dy dx =− E RN \E Z Z i h + Kδ (x − y)Dxj ui (x)Dxi uj (x) dy dx E RN \E Z Z Kδ (x − y)ui (x)Dxi uj (x)ν j (x) dy dx =− ∂E RN \E Z Z i h Kδ (x − y) Dxj ui (x)Dxi uj (x) + ui (x)Dxj xi uj (x) dy dx. + RN \E

E

Therefore A11 + B11 =

Z

Z

Kδ (x − y)h(x)2 ν(x)ν(y) dy dx ∂E ∂E Z Z i h + Kδ (x − y) Dxi ui (x)uj (x)ν j (x) − ui (x)Dxi uj (x)ν j (x) dy dx ∂E RN \E Z Z i h Kδ (x − y) Dxj ui (x)Dxi uj (x) − div(u)(x)2 dy dx, + E

RN \E

so that A11 + B11 + C1 =

Z

∂E

+

Z

∂E

Z

∂E

Kδ (x − y)h(x)2 ν(x)ν(y) dy dx Z i h Kδ (x − y) Dxi ui (x)uj (x)ν j (x) − ui (x)Dxi uj (x)ν j (x) dy dx. RN \E

But using u = νh and div(ν) = H where H is the mean curvature of ∂E we have Dxi ui (x)uj (x)ν j (x) − ui (x)Dxi uj (x)ν j (x) = h(x)2 H(x) and therefore A11 + B11 + C1 =

Z

∂E

Z

∂E

Kδ (x − y)h(x)2 ν(x)ν(y) dy dx +

Z

∂E

Z

RN \E

Kδ (x − y)h(x)2 H(x).

In a similar way, we have Z

Z

Kδ (x − y)h(y)2 ν(x)ν(y) dy dx ∂E ∂E Z Z i h Kδ (x − y) Dyi ui (y)uj (y)ν j (y) − ui (y)Dyi uj (y)ν j (y) dy dx − Z Z Z ZE ∂E 2 Kδ (x − y)h(y)2 H(y) dy dx. Kδ (x − y)h(y) ν(x)ν(y) dy dx − =

A22 + B22 + C2 =

∂E

∂E

E

∂E


14

Further calculations show that Z Z Kδ (x − y)h(x)h(y) dydx A12 = − ∂E ∂E Z Z − Kδ (x − y)div(u)(y)ui (x)ν i (x) dy dx ∂E RN \E Z Z Kδ (x − y)div(u)(x)ui (y)ν i (y) dy dx + E ∂E Z Z Kδ (x − y)div(u)(x)div(u)(y) dy dx, + RN \E

E

A21 = −

Z

Z

∂E

− +

∂E

Z

∂E

+

Z

Z Z E

Kδ (x − y)h(x)h(y) dydx

RN \E

∂E

Kδ (x − y)div(u)(x)ui (y)ν i (y) dy dx

Z Z E

Kδ (x − y)div(u)(y)uj (x)ν j (x) dy dx

Kδ (x − y)div(u)(x)div(u)(y) dy dx,

RN \E

and B12 + B21 = 2

Z

∂E

−2 −4

Z

RN \E

Z Z

Kδ (x − y)div(u)(y)uj (x)ν j (x) dy dx

ZE Z∂E E

Kδ (x − y)div(u)(x)uj (y)ν j (y) dy dx

RN \E

Kδ (x − y)div(u)(x)div(u)(y) dy dx,

so that A12 + A21 + B12 + B21 + C3 = −2

Z

∂E

Z

∂E

Kδ (x − y)h(x)h(y) dydx.

Therefore Z Z d2 P er (E , Ω) = 2 Kδ (x − y)h(x)2 (ν(x)ν(y) − 1) dy dx s,δ th dt2 t=0 ∂E ∂E Z Z Kδ (x − y)(h(y) − h(x)) dydx h(x) −2 ∂E Z Z ∂E 2 (χE (y) − χE c (y))Kδ (x − y) dy dx. h(x) H(x) − ∂E

Taking the limit as δ → 0 we find (4.1).

RN

Proof of proposition 4.2. Let νt (x) denote the unit normal vector to ∂Et at x ∈ ∂Et pointing out of Et . Note that ν(x) = ν0 (x). Let Lt be the half space defined by Lt = {x : hx − pt , νt (pt )i > 0}. Then Z χEt (x) − χLt (x) − χE c (x) + χLct (x) s HΣth (pt ) = dx (4.5) |x − pt |N +s N R


15

since the function 1 − 2χLt has zero principal value. Note that the integral in (4.5) is well defined and Z χEt (x) − χLt (x) dx. HΣs th (pt ) = 2 |x − pt |N +s N R

For δ > 0 let η ∈ C ∞ (RN ) be a radially symmetric cut-off function with η(x) = 1 for |x| ≥ 2, η(x) = 0 for |x| ≤ 1. Define ηδ (x) = η(x/δ) and write Z χEt (x) − χLt (x) dx = fδ (t) + gδ (t) |x − pt |N +s RN where

fδ (t) =

Z

RN

χEt (x) − χLt (x) ηδ (x − pt ) dx |x − pt |N +s

and gδ (t) is the rest. Then it is direct that fδ is differentiable and Z h(x) ′ fδ (0) = η (x − p) N +s δ ∂E |x − p| Z h(p)hν(p), ν(p)i − hx − p, ∂νt∂t(pt ) |t=0 i − ηδ (x − p) |x − p|N +s ∂L0 Z χE (x) − χL0 (x) + (N + s)h(p) hx − p, ν(p)iηδ (x − p)dx |x − p|N +s+2 N R Z χE (x) − χL0 (x) − h(p) h∇ηδ (x − p), ν(p)idx. |x − p|N +s N R

We integrate the third term by parts Z χE (x) − χL0 (x) (N + s) hx − p, ν(p)iηδ (x − p)dx |x − p|N +s+2 N Z R 1 , ν(p)iηδ (x − p)dx (χE (x) − χL0 (x))h∇ =− |x − p|N +s N ZR Z hν(p), ν(p)i hν(x), ν(p)i =− η (x − p) + η (x − p) N +s δ N +s δ |x − p| ∂E ∂L0 |x − p| Z χE(x) − χL0 (x) + h∇ηδ (x − p), ν(p)idx. |x − p|N +s RN Since ηδ is radially symmetric, Z hx − p, ∂νt∂t(pt ) |t=0 i ηδ (x − p) dx = 0 |x − p|N +s ∂L0 and then

fδ′ (0) =

Z

∂E

h(x) ηδ (x − p)dx − h(p) |x − p|N +s

Z

∂E

hν(x), ν(p)i ηδ (x − p)dx, |x − p|N +s

which we write as Z Z h(x) − h(p) 1 − hν(x), ν(p)i fδ′ (0) = η (x − p) dx + h(p) ηδ (x − p) dx. δ N +s |x − p|N +s ∂E |x − p| ∂E

We claim that gδ′ (t) → 0 as δ → 0, uniformly for t in a neighborhood of 0. Indeed, in a neighborhood of pt we can represent ∂Et as a graph of a function Gt over Lt ∩


16

B(pt , 2δ), with Gt defined in a neighborhood of 0 in RN −1 , Gt (0) = 0, ∇y′ Gt (0) = 0 and smooth in all its variables (we write y ′ ∈ RN −1 ). Then gδ (t) becomes Z Z Gt (y′ ) 1 gδ (t) = (1 − ηδ (y ′ , yN ))dyN dy ′ 2 ) N2+s (|y ′ |2 + yN |y ′ |