Nonlocal Delaunay surfaces

0.28 MB·English

by  Juan Dávila

NONLOCAL DELAUNAY SURFACES
JUAN DA´VILA, MANUEL DEL PINO, SERENA DIPIERRO, AND ENRICO VALDINOCI
Abstract. Weconstructcodimension1surfacesofanydimensionthatminimizeaperiodicnonlocalperime-
ter functional among surfaces that are periodic, cylindrically symmetric and decreasing.
These surfaces may be seen as a nonlocal analogue of the classical Delaunay surfaces (onduloids). For
small volume, most of their mass tends to be concentrated in a periodic array and the surfaces are close to
a periodic array of balls (in fact, we give explicit quantitative bounds on these facts).
5
1
0
2
Notation
t
c
O Most of the notation used in this paper is completely standard. For the convenience of the reader, to
avoid ambiguities, we state it clearly from the beginning. The standard Euclidean basis of Rn is denoted
3
by e , ,e (so that, in particular, e = (1,0, ,0)). If E Rn and v Rn, we use the notation
1 n 1
] ··· ··· ⊆ ∈
P E +v := p+v, p E .
A { ∈ }
Also, E is the Lebesgue measure of E. The (n 1)-dimensional Hausdorff measure is denoted by H n 1.
. −
h | | −
The notation ∆ will be used for the symmetric difference, i.e. E∆F := (E F) (F E). We denote
t
a \ ∪ \
by χ the characteristic function of a set E, i.e.
m E
[ 1 if x E,
χ (x) := ∈
E 0 if x E.
2
(cid:26) 6∈
v
Also, throughout the paper, the world “decreasing” stands simply for “non increasing”.
9
5
4 1. Introduction
7
0 The main goal of this paper is to construct a nonlocal analogue of the classical Delaunay surfaces
.
1(see [7]), i.e. surfaces that minimize a fractional perimeter functional among cylindrically decreasing
0
symmetric competitors that are periodic in a given direction. The notion of perimeter that we take into
5
1account is a periodic functional of fractional type, whose critical points are related to axially symmetric
:
vobjects.
i We also study the main geometric properties of the minimizers, such as dislocation of mass and closeness
X
to periodic array of balls.
r
a For this scope, we will introduce a new fractional perimeter functional that takes into account the
periodicity of the surfaces and we develop a fine analysis of the functional in order to obtain suitable
compactness properties. The setting we work in is the following. We consider a fractional parameter s
(0,1). We use coordinates x = (x ,x) R Rn 1 = Rn, with n > 2, and deal with the slab ∈
1 ′ −
∈ ×
S := [ 1/2, 1/2] Rn 1.
−
− ×
We consider the kernel K : Rn (Z Rn 1) R,
−
\ × →
1
K(x) :=
x+ke n+s
k Z | 1|
X∈
and, given a set E Rn, we define
⊆
dxdy
P (E) := K(x y)dxdy = .
S − x y +ke n+s
ZE∩SZS\E ZE∩SZS\EXk∈Z | − 1|
This fractional functional is related to, but quite different from, the nonlocal perimeter introduced in [4]
(namely, it shares with it some nonlocal features, but it has different scaling behaviors and periodicity
1
2 J. DA´VILA,M. DEL PINO,S. DIPIERRO,ANDE. VALDINOCI
properties). More precisely, on the one hand, the functional studied here may be considered as a periodic
version (in the horizontal direction) of the fractional perimeter in [4]. On the other hand, the kernel that
we consider is non-standard, since it has different scaling properties in the different coordinate directions.
We consider the class of our competitors K , that is given by the sets F S of the form
⊆
F = (x ,x) S with x 6 f(x ) ,
1 ′ ′ 1
∈ | |
for a given even function f : [ 1/2, 1/2] [0,+ ] that is decreasing in [0, 1/2].
(cid:8) (cid:9)
In this setting, we prove th−e existence →of volum∞e constrained minimizers of P in K :
S
Theorem 1. For any µ > 0 there exists a minimizer for P in K with volume constraint equal to µ.
S
More explicitly, for any µ > 0 there exists a set F K such that F = µ and, for any F K such
that F = µ, we have that P (F ) 6 P (F). ∗ ∈ | ∗| ∈
S S
| | ∗
Recently, in the literature, there has been an intense effort towards the construction of geometric object
of nonlocal nature that extend classical (i.e. local) ones, see e.g. [6, 9, 8]. In some cases, the nonlocal
objects inherit strong geometric properties from the classical case, but also important differences arise. In
our setting, we think it is an interesting problem to determine whether cylinders are minimizers for large
volume.
As for small volumes, the next result points out (in a quantitative way) that in this case the minimizing
set does not put a considerable proportion of mass close to the boundary of the slab (in particular, it is
“far from being a cylinder”):
Theorem 2. Let
F = (x ,x) S s.t. x 6 f(x )
1 ′ ′ 1
∗ ∈ | |
be a minimizer with volume constraint µn> 0, as given in Theorem 1.oThen
f(1/4) s
(1.1) 6 Cµn2(n 1),
1 −
µn 1
−
for some C > 1. In particular, for any δ (0,1), if µ (0,C 1δn2), we have that
− s
∈ ∈
F x > 1/4
(1.2) ∗ ∩{| 1| } 6 δ
F
(cid:12) (cid:12)
(cid:12) | ∗| (cid:12)
for a suitable constant C > 1.
In case of small volumes, we also show that minimizers are close to balls. The notion of closeness will
be measured by the so-called Fraenkel asymmetry (or symmetric deficit) of a set E, which is defined as
E∆B
Def(E) := inf | |,
E
| |
where the infimum is taken over every ball B Rn with B = E . Roughly speaking, the Fraenkel asym-
⊂ | | | |
metry measures the L1 distance of E from being a ball of the same volume (the ball may be conveniently
translated in order to cover the set E as much as possible, and the quantity above is normalized with
respect to the volume in order to be scale invariant). In this setting we have:
Theorem 3. Let F S be a minimizer according to Theorem 1, with volume constraint µ. Then, if µ is
small enough, F is∗c⊆lose to a ball.
More precisely∗, for any µ (0,1), we have that
∈
n2 s2
Def(F ) 6 Cµ2n2(−n 1).
−
∗
We observe that the control in the sense of deficit obtained in Theorem 3 may lead to further uniform
(though perhaps less explicit) asymptotic bounds, also in the C1-sense, by exploiting suitable density
estimates and approximation results, see e.g. Corollary 3.6 in [9] and Theorem 6.1 in [8].
We point out that, after the present paper was completed and submitted for publication, the very
interesting article [3] appeared, dealing with surfaces of constant fractional mean curvature (in the sense
NONLOCAL DELAUNAY SURFACES 3
of [4, 1]). This paper also provides very fascinating fractional counterparts of the Delaunay surfaces (in
a different setting than the one considered here, and in the planar case, with an announcement of the
n-dimensional results to come).
The rest of the paper is organized as follows. In Section 2 we study the decay properties of the kernel.
Then, in Section 3 we give a detailed comparison between our functional and the one in [4] (this is not
only interesting for seeing similarities and differences with the existing literature, but it is also useful for
constructing competitors and deriving estimates).
Asamatteroffact, theproofofTheorem1alsorequiresacarefulenergyanalysisandad-hoccompactness
arguments in order to use the direct minimization method: these arguments are collected in Sections 4
and 5. With this, all the preliminary work will be completed, and we will be able to prove Theorems 1, 2
and 3 in Sections 6, 7 and 8, respectively.
The paper ends with two appendices. First, in Appendix A, we show that the limit as s 1 of our
ր
fractional functional converges to the classical “periodic” perimeter (i.e. to the perimeter on the cylinder
obtained by identifying the “sides” of the slab S). Then, in Appendix B, we remark that the assumption of
cylindrical symmetry for the competitors in K can be relaxed (in the sense that our fractional functional
decreases under cylindrical rearrangements).
2. Kernel decay
First, we point out that our functional is compatible with the periodic structure in the horizontal
direction. For this, if F S, we define the periodic extension of F as
⊆
F := (F +ke ),
per 1
k Z
[∈
and we have:
Lemma 4. For any τ R, it holds that P (F +τe ) = P (F ).
S per 1 S per
∈
Proof. The function x χ (x ,x) is 1-periodic, and so is x K(x ,x), for any fixed x Rn 1.
Therefore, for any τ 1R7→, Fper 1 ′ 1 7→ 1 ′ ′ ∈ −
∈
dx1 dy1χFper+τe1(x1,x′)χRn\(Fper+τe1)(y1,y′)K(x−y)
Z[ 1/2,1/2] Z[ 1/2,1/2]
− −
= dx dy χ (x ,x)χ (y ,y )K(x y)
1 1 Fper 1 ′ Rn\Fper 1 ′ −
Z[ 1/2,1/2] τ Z[ 1/2,1/2] τ
− − − −
= dx1 dy1χFper(x1,x′)χRn\Fper(y1,y′)K(x−y).
Z[ 1/2,1/2] Z[ 1/2,1/2]
− −
Thus the desired result follows by integrating over x and y . (cid:3)
′ ′
Now we prove a useful decay estimate on our kernel. We remark that the scaling properties of our kernel
are quite different from the ones of many nonlocal problems that have been studied in the literature: as a
matter of fact, the kernel that we study is not homogeneous and it has quite different singular behaviors
locally and at infinity. Indeed, close to the origin the dominant term is of the order of x n s, but at
− −
| |
infinity the x direction “averages out”, as detailed in the following result:
1
Lemma 5. For any x = (x ,x) Rn with x > 1, we have that K(x) 6 C x 1 n s, for some C > 0.
1 ′ ′ ′ − −
∈ | | | |
Proof. Fix x = (x ,x) Rn with x > 1. Let a := x x +1 and b := 3 x x . Let also y be
1 ′ ′ ′ 1 ′ 1
∈ | | ± ±| |− ± ± | |− ±
the integer part of b . We observe that
±
b = 3 x x > x +2 x = a +1.
+ ′ 1 ′ 1 +
| |− | | −
This says that at least one integer lies in the segment [a ,b ] and so y > a . Therefore
+ + + +
y +x 1 > a +x 1 = x .
+ 1 + 1 ′
− − | |
4 J. DA´VILA,M. DEL PINO,S. DIPIERRO,ANDE. VALDINOCI
Moreover
y +x +1 6 b +x +1 6 x 1.
1 1 ′
− − −| |−
Consequently
1 1 1
6 +
x +k n+s (x +k)n+s ( x k)n+s
k Z | 1 | k Z 1 k Z − 1 −
k∈(−∞,bX−∈]∪[b+,+∞) kX>∈y+ k6Xy∈−+1
1 1
= +
(x +k)n+s ( x +k)n+s
k Z 1 k Z − 1
kX>∈y+ k>X−∈y−−1
x1+k dt x1+k dt
−
6 +
tn+s tn+s
kXk>∈yZ+ Zx1+k−1 k>X−k∈y−Z−1 Z−x1+k−1
+ dt + dt
∞ ∞
= +
tn+s tn+s
Zx1+y+ 1 Z x1 y 2
− − − −−
+ dt + dt
∞ ∞
6 +
tn+s tn+s
Z|x′| Z|x′|
C
= .
x n+s 1
′ −
| |
This says that
1 C
(2.1) 6 .
x +k n+s x n+s 1
k Z | 1 | | ′| −
k6∈(−3|x′|X−x∈1,3|x′|−x1)
Now, weobservethattheinterval[ 3 x x , 3 x x ]haslength6 x andsoitcontainsatmost6 x +1 6
′ 1 ′ 1 ′ ′
− | |− | |− | | | |
7 x integers. This implies that
′
| |
1 7
(2.2) 6 .
x n+s x n+s 1
k Z | ′| | ′| −
k∈[−3|x′|X−x∈1,3|x′|−x1]
Moreover,
n+s
x+ke n+s = x +k 2 + x 2 2 > max x +k n+s, x n+s .
1 1 ′ 1 ′
| | | | | | | | | |
Thus, recalling (2.1) and (2.2), we(cid:16)conclude that (cid:17) (cid:8) (cid:9)
1 1
K(x) 6 +
x+ke n+s x+ke n+s
k Z | 1| k Z | 1|
k6∈(−3|x′|X−x∈1,3|x′|−x1) k∈[−3|x′|X−x∈1,3|x′|−x1]
1 1
6 +
x +k n+s x n+s
k Z | 1 | k Z | ′|
k6∈(−3|x′|X−x∈1,3|x′|−x1) k∈[−3|x′|X−x∈1,3|x′|−x1]
C +7
6 ,
x n+s 1
′ −
| |
which gives the desired claim up to renaming C. (cid:3)
Corollary 6. Fix M N. We define
∈
(2.3) h (x) := min M, K(x) .
M
{ }
Then, for any x Rn,
∈
h (x) 6 C min 1, x 1 n s ,
M M ′ − −
{ | | }
for some C > 0 possibly depending on M.
M
NONLOCAL DELAUNAY SURFACES 5
Proof. If x > 1, we use Lemma 5 to see that
′
| |
min 1, x 1 n s = x 1 n s > C 1K(x) > C 1h (x).
′ − − ′ − − − − M
{ | | } | |
On the other hand, if x < 1, we have that
′
| |
min 1, x 1 n s = 1 > M 1h (x).
′ − − − M
{ | | }
Combining these two estimates we obtain the desired result. (cid:3)
3. Relation with the fractional perimeter
The aim of this section is to point out the relation between our functional and the fractional perime-
ter Per introduced in [4]. That is, we set
s
dxdy
Per (F) :=
s x y n+s
ZF ZRn\F | − |
and we show that: on the one hand, our functional is always below the fractional perimeter Per , on the
s
other hand, our functional is always above the fractional perimeter Per , up to a correction that depends
s
on higher order volume terms, and on a volume term coming from the boundary of the slab S. The precise
statement goes as follows:
Proposition 7. Let F S. Then
⊆
(3.1) P (F) 6 Per (F).
S s
More precisely,
dxdy
(3.2) Per (F) P (F) = .
s S
− x y +ke n+s
k Z 0 ZF ZF | − 1|
∈X\{ }
In addition, if we set
F := F x [1/4, 1/2] ,
1
∩{ ∈ }
F := (F +e ) x [1/2, 3/4]
(3.3) 1 1
e ∩{ ∈ }
dxdy
and Π (F) := ,
bS e b x y n+s
ZF ZF | − |
we have that
(3.4) Per (F) 6 P (F)+C F 2 +Π (F) ,
s S S
| |
for some C > 0. (cid:16) (cid:17)
Proof. We use the change of variable y = y +ke to see that
1
dxdy dxdy dxdy
P (F) = = = .
S x y +kee n+s x y n+s x y n+s
ZF ZS\F Xk∈Z | − 1| ZF Z(S\F)+ke1Xk∈Z | − |e ZF Z(S\F)per | − |e
We observe that (S F) Rn F, so we obtain that
per
\ ⊆ \ e e
dxdy dxdy
P (F) 6 6 = Per (F).
S x y n+s x y n+s s
ZF Z(S\F)per | − | ZF ZRn\F | − |
This establishes (3.1). More generally, we see that
(Rn F) (S F) = (F +ke ),
per 1
\ \ \
k Z 0
(cid:0) (cid:1) ∈[\{ }
therefore, with another change of variable, we have
dxdy dxdy
(3.5) Per (F) P (F) = = .
s − S x y n+s x y +ke n+s
k Z 0 ZF ZF+ke1 | − | k Z 0 ZF ZF | − 1|
∈X\{ } ∈X\{ }
6 J. DA´VILA,M. DEL PINO,S. DIPIERRO,ANDE. VALDINOCI
This proves (3.2). Now we observe that, if k > 2 and x, y S, then
| | ∈
k
x y +ke > x y +k > k x y > k 1 > | |,
1 1 1 1 1
| − | | − | | |−| − | | |− 2
therefore
dxdy dxdy
(3.6) 6 6 C F 2,
x y +ke n+s ( k /2)n+s | |
k Z 0 ZF ZF | − 1| k Z 0 ZF ZF | |
∈Xk\>{2} ∈Xk\>{2}
| | | |
for some C > 0. Moreover,
dxdy dxdy
(3.7) χ ( x y ) 6 6 C F 2,
[1/4,+∞) | 1 − 1| x y n+s (1/4)n+s | |
ZF ZF+e1 | − | ZF ZF+e1
for some C > 0. Also, if x S, y S +e and x y 6 1/4, we have that
1 1 1
∈ ∈ | − |
1 1 1
x > y x y > +1 =
1 1 1 1
−| − | −2 − 4 4
1 1 1
and y 1 6 y x +x 1 6 + 1 = .
1 1 1 1
− | − | − 4 2 − −4
As a consequence, if x F S and y F + e S + e , with x y 6 1/4, we have that x F
1 1 1
∈ ⊆ ∈ ⊆ | − | ∈
and y F, where the notation in (3.3) is here in use, therefore
∈
e
dxdy
χ ( x y ) 6 Π (F).
b [0,1/4] | 1 − 1| x y n+s S
ZF ZF+e1 | − |
This and (3.7) give that
dxdy
(3.8) 6 C F 2+Π (F) .
x y n+s | | S
ZF ZF+e1 | − | (cid:16) (cid:17)
Thus, using the change of variable x¯ := x+e and y¯:= y +e , we also have that
1 1
dxdy dxdy
(3.9) = 6 C F 2 +Π (F) .
x y n+s x y n+s | | S
ZF ZF−e1 | − | ZF+e1ZF | − | (cid:16) (cid:17)
Putting together (3.8) and (3.9) we obtain
dxdy dxdy dxdy
= + 6 C F 2 +Π (F) .
x y +ke n+s x y n+s x y n+s | | S
k∈XkZ\6{10} ZF ZF | − 1| ZF ZF+e1 | − | ZF ZF−e1 | − | (cid:16) (cid:17)
| |
This and (3.6) imply that
dxdy
6 C F 2 +Π (F) .
x y +ke n+s | | S
k∈XZ\{0}ZF ZF | − 1| (cid:16) (cid:17)
Recalling (3.5), we see that this ends the proof of (3.4). (cid:3)
As a consequence of our preliminary computations, we obtain that cylinders have finite energy:
Corollary 8. The functional attains a finite value on cylinders. Namely, for any R > 0, let C :=
(x ,x) R Rn 1 s.t. x 6 R . Then P (C S) < + .
1 ′ − ′ S
{ ∈ × | | } ∩ ∞
Proof. We take a set C with smooth boundary and contained in [ 1, 1] Rn 1 such that C S = C S.
−
− × ∩ ∩
Then, we use (3.1) and we obtain that
e e
+ > Per (C) > P (C S) = P (C S),
s S S
∞ ∩ ∩
as desired. (cid:3)
e e
Next result computes the term Π in (3.4) in the special case of small cylinders (this will play a role in
S
the proof of Theorem 3).
NONLOCAL DELAUNAY SURFACES 7
Lemma 9. For any r (0, 1/4), let C := (x ,x) S s.t. x 6 r . Then Π (C) 6 Crn s.
1 ′ ′ S −
∈ { ∈ | | }
Proof. We first translate in the first coordinate and then change variable X := x/r and Y := y/r, so that
we obtain
1/2 3/4 1
Π (C) = dx dy dx dy
S 1 1 ′ ′ x y n+s
Z1/4 Z1/2 Z|x′|6r Z|y′|6r | − |
0 1/4 1
(3.10) = dx dy dx dy
1 1 ′ ′ x y n+s
Z−1/4 Z0 Z|x′|6r Z|y′|6r | − |
0 1/(4r) 1
= rn s dX dY dX dY .
− 1 1 ′ ′ X Y n+s
Z−1/(4r) Z0 Z|X′|61 Z|Y′|61 | − |
Now we take a bounded set C⋆ [ 1,0] Rn 1 with smooth boundary that contains (x ,x)
− 1 ′
Rn s.t. x [ 1,0] and x 6 1 . T⊂hen−we ha×ve that { ∈
1 ′
∈ − | | }
0 1 1 dXdY
dX dY dX dY 6 6 Per (C⋆) 6 C,
1 1 ′ ′ X Y n+s X Y n+s s
Z−1 Z0 Z|X′|61 Z|Y′|61 | − | ZC⋆ ZRn\C⋆ | − |
for some C > 0, thus (3.10) becomes
1 1/(4r) 1
Π (C) 6 Crn s 1+ − dX dY dX dY
S − 1 1 ′ ′ X Y n+s
(3.11) Z−1/(4r) Z0 Z|X′|61 Z|Y′|61 | − |
0 1/(4r) 1
+ dX dY dX dY .
1 1 ′ ′ X Y n+s
Z−1/(4r) Z1 Z|X′|61 Z|Y′|61 | − | !
Notice that we can change variable (x,y) := ( Y, X) and see that
− −
0 1/(4r) 1
dX dY dX dY
1 1 ′ ′ X Y n+s
Z−1/(4r) Z1 Z|X′|61 Z|Y′|61 | − |
1/(4r) 1 1
−
= dy dx dy dx .
1 1 ′ ′ x y n+s
Z0 Z−1/(4r) Z|y′|61 Z|x′|61 | − |
As a consequence, we can write (3.11) as
1 1/(4r) 1
(3.12) Π (C) 6 Crn s 1+2 − dx dy dx dy .
S − 1 1 ′ ′
x y n+s
Z−1/(4r) Z0 Z|x′|61 Z|y′|61 | − | !
Now we observe that
1 1/(4r) 1
−
dx dy dx dy
1 1 ′ ′ x y n+s
Z−1/(4r) Z0 Z|x′|61 Z|y′|61 | − |
1 1/(4r) 1
−
6 dx dy dx dy
1 1 ′ ′ x y n+s
Z−1/(4r) Z0 Z|x′|61 Z|y′|61 | 1 − 1|
1 1/(4r) 1
−
= C dx dy
1 1 x y n+s
Z−1/(4r) Z0 | 1 − 1|
1
6 C − dx ( x )1 n s
1 1 − −
−
Z 1/(4r)
−
+
6 C ∞ dτ τ1 n s
− −
Z1
6 C.
The desired result thus follows by plugging this estimate into (3.12). (cid:3)
8 J. DA´VILA,M. DEL PINO,S. DIPIERRO,ANDE. VALDINOCI
4. Energy bounds
We consider here an auxiliary energy functional and we prove that the functional P is bounded from
S
below by it. The proof requires a very careful analysis of the different contributions and the result, together
with the one in the subsequent Proposition 11, will play a crucial role for the proof of Theorem 1, since it
will lead to the compactness of the minimizing sequences.
Proposition 10. Let F S. Suppose that there exists an even function f : [ 1/2, 1/2] [0,+ ], with
⊆ − → ∞
f decreasing in [0, 1/2], such that
F := (x ,x) S s.t. x 6 f(x ) .
1 ′ ′ 1
∈ | |
n o
Let ε 6 α [0, 1/2] and suppose that
∗ ∗ ∈
(4.1) f(x ) > 4 for all x [0,ε )
1 1
∈ ∗
and
(4.2) f(x ) > 2f(y ) for all x [0,ε ) and y (α , 1/2].
1 1 1 1
∈ ∗ ∈ ∗
Then
ε 1/2 fn 1(x )
P (F) > C ∗ − 1 dy dx ,
S x y 1+s 1 1
Z0 "Zα | 1 − 1| #
∗
for a suitable constant C > 0.
Proof. We have
P (F) = K(x y)dxdy
S
−
ZF ZS F
\
dxdy
>
x y n+s
ZF ZS\F | − |
1/2 1/2 1
= dx dy dx dy .
1 1 ′ ′ x y n+s
Z−1/2 Z−1/2 Z{|x′|6f(x1)} Z{|y′|>f(y1)} | − |
Now we introduce cylindrical coordinates by writing x = ρθ and y = rω, with θ, ω Sn 2. We obtain
′ ′ −
∈
1/2 1/2 f(x1) + ρn 2rn 2
(4.3) P (F) > dx dy dθ dω dρ ∞dr − − .
S 1 1 n+s
Z−1/2 Z−1/2 ZSn−2 ZSn−2 Z0 Zf(y1) |x1 −y1|2 +|ρθ −rω|2 2
Here and in the sequel, dθ and dω are short notations for dH n−1(θ) and(cid:0) dH n−1(ω), respective(cid:1)ly. Now,
for any θ Sn 2, we consider a rotation R on Sn 2 such that θ = R e . Then, we can rotate ω = R ω,
− θ − θ 2 θ
∈
and obtain that
ρn−2rn−2 ρn−2rn−2 e
dω = dω
n+s n+s
ZSn−2 x1 y1 2 + ρθ rω 2 2 ZSn−2 x1 y1 2 + ρRθe2 rω 2 2
| − | | − | | − | | − |
ρn 2rn 2 ρn 2rn 2
(cid:0) − (cid:1) − (cid:0) − (cid:1) −
= dω = dω .
n+s n+s
ZSn−2 x1 y1 2 + Rθ(ρe2 rω) 2 2 ZSn−2 x1 y1 2 + ρe2 rω 2 2
| − | | − | | − | | − |
Notice that the latter inete(cid:0)gral in now independent of θ(cid:1). Then we canew(cid:0)rite (4.3) as (cid:1)
e e
1/2 1/2 f(x1) + ρn 2rn 2
(4.4) P (F) > C dx dy dω dρ ∞dr − − ,
S 1 1 n+s
Z−1/2 Z−1/2 ZSn−2 Z0 Zf(y1) |x1 −y1|2 +|ρe2 −rω|2 2
for some C > 0. (cid:0) (cid:1)
Now we observe that
ρe rω 2 = ρ2 +r2 2ρrω = ρ r 2+2ρr(1 ω ).
2 2 2
| − | − | − | −
NONLOCAL DELAUNAY SURFACES 9
where ω = ω e is the second component of the vector ω Sn 2 0 Rn 1. Now1 we define ω :=
2 2 − −
· ∈ ⊂ { }×
(ω , ,ω ). Then ω = (0,ω ,ω) and
3 n 2
···
(4.5) ω2 + ω 2 = 1.
2 | |
We also set
9
Sn 2 := ω = (0,ω ,ω) Sn 2 s.t. ω > .
⋆− 2 ∈ − 2 10
(cid:26) (cid:27)
Using (4.5), we see that,
√19
ω 6 Sn 2.
| | 10 ⊆ ⋆−
( )
Also, in Sn 2,
⋆−
2ρr(1 ω2)
2ρr(1 ω ) = − 2 6 2ρr(1 ω2) = 2ρr ω 2.
− 2 1+ω − 2 | |
2
Therefore, fixed any a > 0, we have
dω dω
=
n+s n+s
ZSn−2 a2 + ρe2 rω 2 2 ZSn−2 a2 + ρ r 2 +2ρr(1 ω2) 2
| − | | − | −
dω dω
>(cid:0) (cid:1) (cid:0) > (cid:1)
n+s n+s
ZS⋆n−2 a2 + ρ r 2 +2ρr(1 ω2) 2 ZS⋆n−2 a2 + ρ r 2 +2ρr ω 2 2
| − | − | − | | |
dω
> C (cid:0) (cid:1) , (cid:0) (cid:1)
n+s
Z ω6√19/10 a2 + ρ r 2 +2ρr ω 2 2
{| | } | − | | |
for some C > 0, possibly different(cid:0)from line to line. So(cid:1)we use polar coordinates Rn−2 ω = Rϕ,
∋
with ϕ Sn 3 and obtain from the latter estimate that
−
∈
dω √19/10 Rn 3dR
(4.6) > C − .
n+s n+s
ZSn−2 a2 + ρe2 rω 2 2 Z0 a2 + ρ r 2 +2ρrR2 2
| − | | − |
Now we observe that, for any X, Y > 0, we have that
(cid:0) (cid:1) (cid:0) (cid:1)
√19/10 Rn−3dR X n−2 1 √1109XY tn−3dt
(4.7) = ,
n+s Y Xn+s n+s
Z0 X2 +Y2R2 2 (cid:18) (cid:19) Z0 1+t2 2
where the change of variable R(cid:0) = Xt/Y w(cid:1)as performed. (cid:0) (cid:1)
Now we denote, for any x > 0,
x tn 3dt
−
Γ(x) := .
n+s
Z0 1+t2 2
We observe that if x [0,1] and t [0,x], then 1+t2 6 2 and so
(cid:0) (cid:1)
∈ ∈
x
Γ(x) > C tn 3dt = Cxn 2,
− −
Z0
up to renaming C > 0. Moreover, if x > 1, then
1 tn 3dt
Γ(x) > − > C.
n+s
Z0 1+t2 2
Summarizing, for any x > 0, we have that
(cid:0) (cid:1)
Γ(x) > C min xn 2, 1 .
−
{ }
1For concreteness we suppose in this part that n > 3. In the very special case n = 2, one does not have any compo-
nent (ω3, ,ωn), so she or he can just disregard ω and go directly to (4.10).
···
10 J. DA´VILA,M. DEL PINO,S. DIPIERRO,ANDE. VALDINOCI
Thus, going back to (4.7),
√19/10 Rn−3dR X n−2 1 √19Y
= Γ
n+s Y Xn+s 10X
Z0 X2 +Y2R2 2 (cid:18) (cid:19) !
(4.8)
n 2 n 2 n 2
(cid:0) X − (cid:1)1 Y − 1 X −
> C min 1, = C min , 1 .
Y Xn+s X Xn+s Y
( ) ( )
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
Now we take X := a2 + ρ r 2 and Y := √2ρr and we plug (4.8) into (4.6). In this way we obtain
| − |
n 2
p dω 1 a2 + ρ r 2 −2
(4.9) > C min | − | , 1 .
n+s n+s ρr
ZSn−2 a2 + ρe2 rω 2 2 a2 + ρ r 2 2 ((cid:18) (cid:19) )
| − | | − |
Hence we take a := x y and we insert (4.9) into (4.4), obtaining that
(cid:0) 1 1 (cid:1) (cid:0) (cid:1)
| − |
1/2 1/2 f(x1) +
∞
P (F) > C dx dy dρ dr
S 1 1
Z 1/2 Z 1/2 Z0 Zf(y1)
(4.10) − −
n 2
ρn−2rn−2 x1 y1 2 + ρ r 2 −2
min | − | | − | , 1 .
n+s ρr
x1 y1 2 + ρ r 2 2 ((cid:18) (cid:19) )
| − | | − |
Now we observe that
(cid:0) (cid:1)
1/2 1/2 f(x1) +
∞
dx dy dρ dr
1 1
Z 1/2 Z 1/2 Z0 Zf(y1)
− −
n 2
ρn 2rn 2 x y 2 + ρ r 2 −2
− − 1 1
χ min | − | | − | , 1
{|x1−y1|2+|ρ−r|26ρr} x1 y1 2 + ρ r 2 n+2s ((cid:18) ρr (cid:19) )
| − | | − |
1/2 1/2 f(x(cid:0)1) +∞ (cid:1) ρn−22rn−22
= dx dy dρ drχ .
1 1 x1 y12+ρ r26ρr 1+s
Z 1/2 Z 1/2 Z0 Zf(y1) {| − | | − | } x1 y1 2 + ρ r 2 2
− − | − | | − |
This and (4.10) give that
(cid:0) (cid:1)
1/2 1/2 f(x1) +∞ ρn−22rn−22
(4.11) P (F) > C dx dy dρ drχ .
S 1 1 x1 y12+ρ r26ρr 1+s
Z 1/2 Z 1/2 Z0 Zf(y1) {| − | | − | } x1 y1 2 + ρ r 2 2
− − | − | | − |
Accordingly, we perform the change of variable ρ = x y α and r = x y β, so that (4.11) becomes
1 1 1 (cid:0)1 (cid:1)
| − | | − |
P (F) > C 1/2 dx 1/2 dy |xf1(−x1y)1| dα +∞ dβχ |x1 −y1|n−2−sαn−22βn−22
S 1 1 1+α β26αβ 1+s
Z−1/2 Z−1/2 Z0 Z|xf1(−y1y)1| { | − | } 1+|α−β|2 2
> C ε∗ dx 1/2dy |xf1(−x1y)1| dα +∞ dβχ |x1 −(cid:0) y1|n−2−sαn−2(cid:1)2βn−22 ,
1 1 1+α β26αβ 1+s
Z0 Zα∗ Z109|fx(1x−1y)1| Z|xf1(−y1y)1| { | − | } 1+|α−β|2 2
where ε and α were introduced in (4.1) and (4.2). As a matter of fact, using (4.2), we obtain that, in
(cid:0) (cid:1)
∗ ∗
the domain above,
f(y ) f(x ) 9f(x )
1 6 1 < 1 6 α.
x y 2 x y 10 x y
1 1 1 1 1 1
| − | | − | | − |
As a consequence
P (F) > C ε∗dx 1/2dy |xf1(−x1y)1| dα +∞dβχ |x1 −y1|n−2−sαn−22βn−22
S 1 1 1+α β26αβ 1+s
(4.12) Z0 Zα∗ Z109|fx(1x−1y)1| Zα { | − | } 1+|α−β|2 2
> C ε∗dx 1/2dy |xf1(−x1y)1| dα α+1dβχ |x1(cid:0)−y1|n−2−sαn(cid:1)−22βn−22 .
1 1 1+α β26αβ 1+s
Z0 Zα∗ Z109|fx(1x−1y)1| Zα { | − | } 1+|α−β|2 2
(cid:0) (cid:1)