A note on the existence of H-bubbles via perturbation methods

15 Pages·0.2 MB·English

by  Veronica Felli

A note on the existence of H-bubbles
via perturbation methods
3
0
0
2 Veronica Felli
∗

n
a
S.I.S.S.A. - Via Beirut 2-4
J

0 34014 Trieste, Italy
3 [email protected]
e-mail:

]
P
A
.
h Abstract
t
a
m We study the problem of existence of surfaces in R3 parametrized on the sphere S2
[ with prescribed mean curvature H in the perturbative case, i.e. for H =H0+εH1,
where H is a nonzero constant, H is a C2 function and ε is a small perturbation
3 0 1
v parameter.
5
6
2
Key Words: H-surfaces, nonlinear elliptic systems, perturbative methods.
1
0 MSC classification: 53A10, 35J50, 35B20.
3
0
/
h 1 Introduction
t
a
m
v: In this paper we are interested in the existence of H-bubbles, namely of S2-type para-
Xi metric surfaces in R3 with prescribed mean curvature H. This geometrical problem is
r motivated by some models describing capillarity phenomena and has the following ana-
a
lytical formulation: given a function H C1(R3), find a smooth nonconstant function
∈
ω : R2 R3 which is conformal as a map on S2 and solves the problem
→
∆ω = 2H(ω)ω ω , in R2,
x y
∧
 (P )
H
 ω 2 < + ,
ZR2|∇ | ∞


where ω = ∂ω1, ∂ω2, ∂ω3 , ω = ∂ω1, ∂ω2, ∂ω3 , ∆ω = ω +ω , ω = (ω ,ω ), and
x ∂x ∂x ∂x y ∂y ∂y ∂y xx yy ∇ x y ∧
denotes the e(cid:0)xterior produ(cid:1)ct in R(cid:0)3. (cid:1)
∗Supported by MIUR, national project “Variational Methods and Nonlinear Differential Equations”.
1
2 V. Felli
Brezis and Coron [4] proved that for constant nonzero mean curvature H(u) H the
0
≡
only nonconstant solutions are spheres of radius H −1.
0
| |
While the Plateau and the Dirichlet problems has been largely studied both for H
constant and for H nonconstant (see [3, 4, 10, 12, 13, 14, 15, 16]), problem (P ) in the
H
caseof nonconstantH has beeninvestigated onlyrecently, see [5,6,7]. In[5]Caldiroliand
Musina proved the existence of H-bubbles with minimal energy under the assumptions
that H C1(R3) satisfies
∈
(i) sup H(u+ξ) uu < 1 for some ξ R3,
u∈R3|∇ · | ∈
(ii) H(u) H as u for some H R,
∞ ∞
→ | | → ∞ ∈
4π
(iii) c = inf sup (su) <
H H
u∈Cc1(R2,R3) s>0 E 3H∞2
u6=0
where (u) = 1 u 2+2 Q(u) u u and Q : R3 R3 is any vector field such
EH 2 R2 |∇ | R2 · x∧ y →
that divQ = H. R R
The perturbative method introduced by Ambrosetti and Badiale [1, 2] was used in [7]
to treat the case in which H is a small perturbation of a constant, namely
H(u) = H (u) = H +εH (u),
ε 0 1
where H R 0 , H C2(R3), and ε is a small real parameter. This method allows to
0 1
∈ \{ } ∈
find critical points of a functional f of the type f (u) = f (u) εG(u) in a Banach space
ε ε 0
−
by studying a finite dimensional problem. More precisely, if the unperturbed functional
f has a finite dimensional manifold of critical points Z which satisfies a nondegeneracy
0
condition, it is possible to prove, for ε sufficiently small, the existence of a smooth
| |
function η (z) : Z (T Z)⊥ such that any critical point z¯ Z of the function
ε z
→ ∈
Φ : Z R, Φ (z) = f z +η (z)
ε ε ε ε
→
(cid:0) (cid:1)
gives rise to a critical point u = z¯ + η (z¯) of f , i.e. the perturbed manifold Z :=
ε ε ε ε
z +η (z) : z Z is a natural constraint for f . Furthermore Φ can be expanded as
ε ε ε
{ ∈ }
Φ (z) = b εΓ(z)+o(ε) as ε 0 (1.1)
ε
− →
where b = f (z) and Γ is the Melnikov function defined as the restriction of the pertur-
0
bation G on Z, namely Γ = G . For problem (P ), Γ is given by
Z Hε
(cid:12)
(cid:12)
Γ : R3 R, Γ(p) = H (q)dq.
1
→ Z 1
|p−q|<
|H0|
In [7] Caldiroli and Musina studied the functional Γ giving some existence results in the
perturbativesetting forproblem(P ). They provethatfor ε smallthereexists asmooth
Hε | |
Existence of H-bubbles 3
H -bubble if one of the following conditions holds
ε
1) H has a nondegenerate stationary point and H is large;
1 0
| |
2) maxH (p) < maxH (p) or min H (p) > minH (p)
1 1 1 1
p∈∂K p∈K p∈∂K p∈K
for some nonempty compact set K R3 and H is large;
0
⊂ | |
3) H Lr(R3) for some r [1,2].
1
∈ ∈
They prove that critical points of Γ give rise to solutions to (P ) for ε sufficiently small.
Hε
Precisely the assumption that H is large required in cases 1) and 2) ensures that if H
0 1
is not constant then Γ is not identically constant. If we let this assumption drop, it may
happen that Γ is constant even if H is not. This fact produces some loss of information
1
because the first order expansion (1.1) is not sufficient to deduce the existence of critical
points of Φ from the existence of critical points of Γ. Instead of studying Γ we perform a
ε
direct study of Φ which allows us to prove some new results. In the first one, we assume
ε
that H vanishes at and has bounded gradient, and prove the existence of a solution
1
∞
without branch points. Let us recall that a branch point for asolution ω to (P ) is a point
H
where ω = 0, i.e. a point where the surface parametrized by ω fails to be immersed.
∇
Theorem 1.1 Let H R 0 , H C2(R3) such that
0 1
∈ \{ } ∈
(H1) lim H (p) = 0;
1
|p|→∞
(H2) H L∞(R3,R3).
1
∇ ∈
Let H = H +εH . Then for ε sufficiently small there exists a smooth H -bubble without
ε 0 1 ε
| |
branch points.
With respect to case 1) of [7] we require neither nondegeneracy of critical points of H
1
nor largeness of H . With respect to case 2) we do not assume that H is large; on the
0 0
other hand our assumption (H1) implies 2). Moreover we do not assume any integrability
condition of type 3). With respect to the result proved in [5], we have the same kind of
behavior of H at (see (ii) and assumption (H1)) but we do not need any assumption
1
∞
of type (iii); on the other hand in [5] it is not required that the prescribed curvature is a
small perturbation of a constant.
ThefollowingresultsgivesomeconditionsonH inordertohavetwoorthreesolutions.
1
Theorem 1.2 Let H R 0 , H C2(R3) such that (H1), (H2),
0 1
∈ \{ } ∈
(H3) HessH (p) is positive definite for any p B (0),
1 ∈ 1/|H0|
(H4) H (p) > 0 in B (0),
1 1/|H0|
hold. Then for ε sufficiently small there exist at least three smooth H -bubbles without
ε
| |
branch points.
4 V. Felli
Remark 1.3 If we assume (H1), (H2), and, instead of (H3) (H4), that H (0) > 0
1
−
and HessH (0) is positive definite, then we can prove that for H sufficiently large and
1 0
| |
ε sufficiently small there exist at least three smooth H -bubbles without branch points.
ε
| |
Theorem 1.4 Let H R 0 , H C2(R3) such that (H1) and (H2) hold. Assume
0 1
∈ \{ } ∈
that there exist p ,p R3 such that
1 2
∈
(H5) H (ξ)dξ > 0 and H (ξ)dξ < 0.
1 1
Z Z
B(p1,1/|H0|) B(p2,1/|H0|)
Then for ε sufficiently small there exist at least two smooth H -bubbles without branch
ε
| |
points.
Remark 1.5 If we assume (H1), (H2), and, instead of (H5), that there exist p ,p R3
1 2
∈
such that H (p ) > 0 and H (p ) < 0, then we can prove that for H sufficiently large
1 1 1 2 0
| |
and ε sufficiently small there exist at least two smooth H -bubbles without branch points.
ε
| |
The present paper is organized as follows. In Section 2 we introduce some notation and
recall some known facts whereas Section 3 is devoted to the proof of Theorems 1.1, 1.2,
and 1.4.
Acknowledgments. TheauthorwishestothankProfessorA.AmbrosettiandProfessor
R. Musina for many helpful suggestions.
2 Notation and known facts
In the sequel we will take H = 1; this is not restrictive since we can do the change
0
H (u) = H H (H u). Hence we will always write
1 0 1 0
e
H (u) = 1+εH(u),
ε
where H C2(R3). Let us denote by ω the function ω : R2 S2 defined as
∈ →
2
ω(x,y) = µ(x,y)x,µ(x,y)y,1 µ(x,y) where µ(x,y) = , (x,y) R2.
− 1+x2 +y2 ∈
(cid:0) (cid:1)
Note that ω is a conformal parametrization of the unit sphere and solves
∆ω = 2ω ω on R2
x y
∧
 (2.1)
 ω 2 < + .
ZR2|∇ | ∞


Existence of H-bubbles 5
Problem (2.1) has in fact a family of solutions of the form ω φ+p where p R3 and φ is
◦ ∈
any conformal diffeomorphism ofR2 . Fors (1,+ ), we will set Ls := Ls(S2,R3),
∪{∞} ∈ ∞
where any map v Ls is identified with the map v ω : R2 R3 which satisfies
∈ ◦ →
v ω sµ2 = v s.
ZR2 | ◦ | ZS2| |
We will use the same notation for v and v ω. By W1,s we denote the Sobolev space
◦
W1,s(S2,R3) endowed (according to the above identification) with the norm
1/s 1/s
v W1,s = v sµ2−s + v sµ2 .
k k (cid:20)ZR2|∇ | (cid:21) (cid:20)ZR2| | (cid:21)
If s′ is the conjugate exponent of s, i.e. 1 + 1 = 1, the duality product between W1,s and
s s′
W1,s′ is given by
v,ϕ = v ϕ+ v ϕµ2 for any v W1,s and ϕ W1,s′.
h i ZR2∇ ·∇ ZR2 · ∈ ∈
Let Q be any smooth vector field on R3 such that divQ = H. The energy functional
associated to problem
∆u = 2 1+εH(u) u u , in R2,
x y
∧
 (cid:0) (cid:1) (P )
ε
 u 2 < + ,
ZR2|∇ | ∞


is given by
1
(u) = u 2 +2 (u)+2ε (u), u W1,3,
ε 1 H
E 2 ZR2|∇ | V V ∈
where
(u) = Q(u) u u
H x y
V ZR2 · ∧
has the meaning of an algebraic volume enclosed by the surface parametrized by u with
weight H (it is independent of the choice of Q); in particular
1
(u) = u u u .
1 x y
V 3 ZR2 · ∧
In [7], Caldiroli and Musina studied some regularity properties of on the space W1,3.
H
V
In particular they proved the following properties.
a) ForH C1(R3), thefunctional isofclass C1 onW1,3 andtheFr´echet differential
H
∈ V
of at u W1,3 is given by
H
V ∈
d (u)ϕ = H(u)ϕ u u for any ϕ W1,3 (2.2)
H x y
V ZR2 · ∧ ∈
6 V. Felli
and admits a unique continuous and linear extension on W1,3/2 defined by (2.2).
Moreover for every u W1,3 there exists ′ (u) W1,3 such that
∈ VH ∈
′ (u),ϕ = H(u)ϕ u u for any ϕ W1,3/2. (2.3)
hVH i ZR2 · x ∧ y ∈
b) For H C2(R3), the map ′ : W1,3 W1,3 is of class C1 and
∈ VH →
′′(u) η,ϕ = H(u)ϕ (η u +u η )+ ( H(u) η)ϕ (u u )
hVH · i ZR2 · x ∧ y x ∧ y ZR2 ∇ · · x ∧ y
for any u,η W1,3 and ϕ W1,3/2. (2.4)
∈ ∈
Hence for all u W1,3, ′(u) W1,3 and for any ϕ W1,3/2
∈ Eε ∈ ∈
′(u),ϕ = u ϕ+2 ϕ u u +2ε H(u)ϕ u u .
hEε i ZR2∇ ·∇ ZR2 · x ∧ y ZR2 · x ∧ y
As remarked in [7], critical points of in W1,3 give rise to bounded weak solutions to (P )
ε ε
E
and hence by the regularity theory for H-systems (see [9]) to classical conformal solutions
which are C3,α as maps on S2.
Theunperturbedproblem,i.e. (P )forε = 0,hasa9-dimensionalmanifoldofsolutions
ε
given by
Z = Rω L +p : R SO(3), l > 0, ξ R2, p R3
l,ξ
{ ◦ ∈ ∈ ∈ }
where L z = l(z ξ) (see [11]). In [11] the nondegeneracy condition T Z = ker ′′(u) for
l,ξ − u E0
any u Z (where T Z denotes the tangent space of Z at u) is proved (see also [8]).
u
∈
As observed in [7], in performing the finite dimensional reduction, the dependence on
the 6-dimensional conformal group can be neglected since any H-system is conformally
invariant. Hence we look for critical points of constrained on a three-dimensional
ε
E
manifold Z just depending on the translation variable p R3.
ε
∈
3 Proof of Theorem 1.1
We start by constructing a perturbed manifold which is a natural constraint for .
ε
E
Lemma 3.1 Assume H C2(R3) L∞(R3) and H L∞(R3,R3). Then there exist
∈ ∩ ∇ ∈
ε > 0, C > 0, and a C1 map η : ( ε ,ε ) R3 W1,3 such that for any p R3 and
0 1 0 0
− × → ∈
ε ( ε ,ε )
0 0
∈ −
′ ω +p+η(ε,p) T Z, (3.1)
Eε ∈ ω
η((cid:0)ε,p) (T Z)⊥,(cid:1) (3.2)
ω
∈
η(ε,p) = 0, (3.3)
ZS2
η(ε,p) W1,3 C1 ε . (3.4)
k k ≤ | |
Existence of H-bubbles 7
Moreover if we assume that the limit of H at exists and
∞
lim H(p) = 0 (3.5)
|p|→∞
we have that η(ε,p) converges to 0 in W1,3 as p uniformly with respect to ε < ε .
0
| | → ∞ | |
Proof. Let us define the map
F = (F ,F ) : R R3 W1,3 R6 R3 W1,3 R6 R3
1 2
× × × × → × ×
6
F (ε,p,η,l,α),ϕ = ′(ω +p+η),ϕ l ϕ τ +α ϕ, ϕ W1,3/2
h 1 i hEε i−Xi=1 iZR2 ∇ ·∇ i ·ZS2 ∀ ∈
F (ε,p,η,l,α) = η τ ,..., η τ , η
2 1 6
(cid:18)ZR2∇ ·∇ ZR2 ∇ ·∇ ZS2 (cid:19)
where τ ,...,τ are chosen in T Z such that
1 6 ω
τ τ = δ and τ = 0 i,j = 1,...,6
i j ij i
ZR2∇ ·∇ ZS2
so that T Z is spanned by τ ,...,τ ,e ,e ,e . It has been proved by Caldiroli and
ω 1 6 1 2 3
Musina [7] that F is of class C1 and that the linear continuous operator
: W1,3 R6 R3 W1,3 R6 R3
L × × → × ×
∂F
= (0,p,0,0,0)
L ∂(η,l,α)
i.e.
6
(v,µ,β),ϕ = ′′(ω) v,ϕ µ ϕ τ β ϕ ϕ W1,3/2
hL1 i hE0 · i−Xi=1 iZR2 ∇ · i − ZS2 ∀ ∈
(v,µ,β) = v τ ,..., v τ , v
2 1 6
L (cid:18)ZR2∇ ·∇ ZR2 ∇ ·∇ ZS2 (cid:19)
is invertible. Caldiroli and Musina applied the Implicit Function Theorem to solve the
equation F(ε,p,η,l,α) = 0 locally with respect to the variables ε,p, thus finding a C1-
functionη onaneighborhood( ε ,ε ) B R R3 satisfying (3.1), (3.2), and(3.3). We
0 0 R
− × ⊂ ×
will use instead the Contraction Mapping Theorem, which allows to prove the existence of
such a function η globally on R3, thanks to the fact that the operator does not depend
L
on p and hence it is invertible uniformly with respect to p R3.
∈
We have that F(ε,p,η,l,α) = 0 if and only if (η,l,α) is a fixed point of the map T
ε,p
defined as
T (η,l,α) = −1F(ε,p,η,l,α)+(η,l,α).
ε,p
−L
8 V. Felli
To prove the existence of η satisfying (3.1), (3.2), and (3.3), it is enough to prove that
T is a contraction in some ball B (0) with ρ = ρ(ε) > 0 independent of p, whereas the
ε,p ρ
regularity of η(ε,p) follows from the Implicit Function Theorem.
We have that if η W1,3 ρ
k k ≤
Tε,p(η,l,α) W1,3×R6×R3
k k
≤ C2kF(ε,p,η,l,α)−L(η,l,α)kW1,3×R6×R3 ≤ C2kEε′(ω +p+η)−E0′′(ω)ηkW1,3
≤ C2 kE0′(ω +η)−E0′′(ω)ηkW1,3 +2|ε|kVH′ (ω +p+η)kW1,3
(cid:0) 1 (cid:1)
≤ C2(cid:18)Z kE0′′(ω +tη)−E0′′(ω)kW1,3/2dtkηkW1,3 +2|ε|kVH′ (ω +p+η)kW1,3(cid:19)
0
≤ C2ρ sup kE0′′(ω +η)−E0′′(ω)kW1,3/2 +2C2|ε| sup kVH′ (ω +p+η)kW1,3
kηkW1,3≤ρ kηkW1,3≤ρ
(3.6)
where C2 = −1 L(W1,3×R6×R3). For (η1,l1,α1), (η2,l2,α2) Bρ(0) W1,3 R6 R3 we
kL k ∈ ⊂ × ×
have
Tε,p(η1,l1,α1) Tε,p(η2,l2,α2) W1,3×R6×R3
k − k
C η η
2 1 2 W1,3
k − k
kEε′(ω +p+η1)−Eε′(ω +p+η2)−E0′′(ω)(η1 −η2)kW1,3
≤ C2 η1 η2 W1,3
k − k
1
′′(ω +p+η +t(η η )) ′′(ω) dt
≤ Z kEε 2 1 − 2 −E0 kW1,3/2
0
1
′′(ω +p+η +t(η η )) ′′(ω) dt
≤ Z kE0 2 1 − 2 −E0 kW1,3/2
0
1
+2 ε ′′(ω +p+η +t(η η )) dt
| |Z kVH 2 1 − 2 kW1,3/2
0
sup ′′(ω +η) ′′(ω) +2 ε sup ′′(ω +p+η) .
≤ kE0 −E0 kW1,3/2 | | kVH kW1,3/2
kηkW1,3≤3ρ kηkW1,3≤3ρ
From (2.3), (2.4), and the H¨older inequality it follows that there exists a positive constant
C such that for any η W1,3, p R3
3
∈ ∈
2/3
kVH′ (ω +p+η)kW1,3 ≤ C3(cid:20)(cid:18)ZR2 |H(ω+p+η)|3/2|∇ω|3µ−1(cid:19) +kηk2W1,3(cid:21) (3.7)
1/2
′′(ω +p+η) C H(ω+p+η) 2 (ω +η) 2
kVH kW1,3/2 ≤ 3(cid:20)(cid:18)ZR2 | | |∇ | (cid:19)
2/3
+ H(ω+p+η) 3/2 (ω +η) 3µ−1 . (3.8)
(cid:18)ZR2 |∇ | |∇ | (cid:19) (cid:21)
Existence of H-bubbles 9
Choosing ρ > 0 such that
0
1
C sup ′′(ω +η) ′′(ω) <
2 E0 −E0 W1,3/2 2
kηkW1,3≤3ρ0(cid:13) (cid:13)
(cid:13) (cid:13)
and ε > 0 such that
0
1
8C2C3ε0kHkL∞(R3)kωk2W1,3 < min(cid:26)1,ρ0, 8C C ε (cid:27), (3.9)
2 3 0
ρ
sup VH′ (ω +p+η)kW1,3 < 6ε 0C , (3.10)
kηkW1,3≤ρ0(cid:13) 0 2
p∈R3 (cid:13)
1
sup ′′(ω +p+η) < , (3.11)
VH kW1,3/2 8ε C
kηkW1,3≤3ρ0(cid:13) 0 2
p∈R3 (cid:13)
we obtain that T maps the ball B (0) into itself for any ε < ε , p R3, and is a
ε,p ρ0 | | 0 ∈
contraction there. Hence it has a unique fixed point η(ε,p),l(ε,p),α(ε,p) B (0).
∈ ρ0
From (3.6) we have that the following property holds (cid:0) (cid:1)
( ) T maps a ball B (0) W1,3 R6 R3 into itself whenever ρ ρ and
ε,p ρ 0
∗ ⊂ × × ≤
ρ > 4|ε|C2 sup VH′ (ω +p+η)kW1,3.
kηkW1,3≤ρ(cid:13)
(cid:13)
In particular let us set
ρε = 5|ε|C2 sup VH′ (ω +p+η)kW1,3. (3.12)
kηkW1,3≤ρ0(cid:13)
p∈R3 (cid:13)
In view of (3.10) and (3.12), we have that for any ε < ε and for any p R3
0
| | ∈
ρε ≤ ρ0 and ρε > 4|ε|C2 sup VH′ (ω +p+η)kW1,3
kηkW1,3≤ρε(cid:13)
(cid:13)
so that, due to ( ), T maps B (0) into itself. From the uniqueness of the fixed point
∗ ε,p ρε
we have that for any ε < ε and p R3
0
| | ∈
(η(ε,p),l(ε,p),α(ε,p)) W1,3×R6×R3 ρε C1 ε (3.13)
k k ≤ ≤ | |
for some positive constant C1 independent of p and hence η(ε,p) W1,3 ρε C1 ε thus
k k ≤ ≤ | |
proving (3.4). Assume now (3.5) and set for any p R3
∈
2/3
ρ = 8C C ε sup H(q) 3/2 ω 3µ−1
p 2 3 0
(cid:18)ZR2|q−p|≤1+C0| | |∇ | (cid:19)
10 V. Felli
whereC0 isapositiveconstantsuchthat u L∞ C0 u W1,3 foranyu W1,3. From(3.9)
k k ≤ k k ∈
we have that
1
ρ < min 1,ρ , .
p (cid:26) 0 8C C ε (cid:27)
2 3 0
Hence, due to (3.7), we have that for ε < ε0 and η W1,3 ρp
| | k k ≤
4|ε|C2 VH′ (ω +p+η)kW1,3
(cid:13) 2/3
(cid:13)
4ε C C sup H(q) 3/2 ω 3µ−1 +4ε C C ρ2 < ρ .
≤ 0 2 3(cid:18)ZR2|q−p|≤1+C0| | |∇ | (cid:19) 0 2 3 p p
From ( ) and the uniqueness of the fixed point, we deduce that η(ε,p) W1,3 ρp for any
∗ k k ≤
ε < ε and p R3. On the other hand, since H vanishes at , by the definition of ρ
0 p
| | ∈ ∞
we have that ρ 0 as p , hence
p
→ | | → ∞
lim η(ε,p) = 0 in W1,3 uniformly for ε < ε .
0
|p|→∞ | |
2
The proof of Lemma 3.1 is now complete.
Remark 3.2 The map η given in Lemma 3.1 satisfies
6
′(ω +p+η(ε,p)),ϕ l (ε,p) ϕ τ +α(ε,p) ϕ, ϕ W1,3/2
hEε i−Xi=1 i ZR2∇ ·∇ i ·ZS2 ∀ ∈
where η(ε,p),l(ε,p),α(ε,p) B (0) W1,3 R6 R3 being ρ given in (3.12), hence
∈ ρε ⊂ × × ε
(cid:0) (cid:1)
(ω +η(ε,p)) ϕ+2 ϕ (ω +η(ε,p)) (ω +η(ε,p))
x y
ZR2∇ ·∇ ZR2 · ∧
+2ε H(ω +p+η(ε,p))ϕ (ω +η(ε,p)) (ω +η(ε,p))
x y
ZR2 · ∧
6
= l (ε,p) ϕ τ α(ε,p) ϕ, ϕ W1,3/2,
i i
Xi=1 ZR2∇ ·∇ − ·ZS2 ∀ ∈
i.e. η(ε,p) satisfies the equation
∆η(ε,p) = F(ε,p)
where
F(ε,p) =2(ω +η(ε,p)) (ω +η(ε,p)) 2ω ω +l(ε,p) ∆τ α(ε,p)µ2
x y x y
∧ − ∧ · −
+2εH(ω+p+η(ε,p))(ω+η(ε,p)) (ω +η(ε,p)) in R2.
x y
∧