The Parabolic Infinite-Laplace Equation in Carnot groups

0.24 MB·English

by  Thomas Bieske

THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT
GROUPS
5 THOMAS BIESKE AND ERIN MARTIN
1
0
Abstract. By employing a Carnot parabolic maximum principle, we show existence-
2
uniquenessofviscositysolutionstoaclassofequationsmodeledontheparabolicinfinite
n
Laplace equation in Carnotgroups. We show stability of solutions within the class and
a
J examine the limit as t goes to infinity.

8
2

]
P
1. Motivation
A
. In Carnot groups, the following theorem has been established.
h
t
a Theorem 1.1. [3,16,5] LetΩ be a bounded domainin a Carnotgroup and let v : ∂Ω → R
m
be a continuous function. Then the Dirichlet problem
[

1 ∆∞u = 0 in Ω
v (cid:26) u = v on ∂Ω
1
8
has a unique viscosity solution u .
1 ∞
7
0 Our goal is to prove a parabolic version of Theorem 1.1 for a class of equations (defined
. in the next section), namely
1
0
5 Conjecture 1.2. Let Ω be a bounded domain in a Carnot group and let T > 0. Let
1
ψ ∈ C(Ω) and let g ∈ C(Ω×[0,T)) Then the Cauchy-Dirichlet problem
:
v
i u −∆h u = 0 in Ω×(0,T),
X t ∞
(1.1)  u(x,0) = ψ(x) on Ω
r
a  u(x,t) = g(x,t) on ∂Ω×(0,T)

has a unique viscosity solution u.
In Sections 2 and 3, we review key properties of Carnot groups and parabolic viscosity
solutions. In Section 4, we prove uniqueness and Section 5 covers existence.
Date: January 15, 2015.
2010 Mathematics Subject Classification. Primary 53C17, 35K65, 35D40; Secondary 35H20, 22E25,
17B70.
Key words and phrases. parabolic p-Laplace equation, viscosity solution, Carnot groups.
The first author was partially supported by a University of South Florida Proposal Enhancement
Grant.
1
2 THOMAS BIESKE ANDERIN MARTIN
2. Calculus on Carnot Groups
We begin by denoting an arbitrary Carnot group in RN by G and its corresponding
Lie Algebra by g. Recall that g is nilpotent and stratified, resulting in the decomposition
g = V ⊕V ⊕···⊕V
1 2 l
for appropriate vector spaces that satisfy the Lie bracket relation [V ,V ] = V . The Lie
1 j 1+j
Algebra g is associated with the group G via the exponential map exp : g → G. Since
this map is a diffeomorphism, we can choose a basis for g so that it is the identity map.
Denote this basis by
X ,X ,...,X ,Y ,Y ,...,Y ,Z ,Z ,...,Z
1 2 n1 1 2 n2 1 2 n3
so that
V = span{X ,X ,...,X }
1 1 2 n1
V = span{Y ,Y ,...,Y }
2 1 2 n2
V ⊕V ⊕···⊕V = span{Z ,Z ,...,Z }.
3 4 l 1 2 n3
We endow g with an inner product h·,·i and related norm k · k so that this basis is
orthonormal. Clearly, the Riemannian dimension of g (and so G) is N = n +n +n .
1 2 3
However, we will also consider the homogeneous dimension of G, denoted Q, which is
given by
l
Q = i·dimV .
i
Xi=1
Before proceeding with the calculus, we recall the group and metric space proper-
ties. Since the exponential map is the identity, the group law is the Campbell-Hausdorff
formula (see, for example, [7]). For our purposes, this formula is given by
1
(2.1) p·q = p+q + [p,q]+R(p,q)
2
where R(p,q) are terms of order 3 or higher. The identity element of G will be denoted
by 0 and called the origin. There is also a natural metric on G, which is the Carnot-
Carath´eodory distance, defined for the points p and q as follows:
1
d (p,q) = inf kγ′(t)kdt
C
Γ Z
0
where the set Γ is the set of all curves γ such that γ(0) = p,γ(1) = q and γ′(t) ∈ V . By
1
Chow’s theorem (see, for example, [2]) any two points can be connected by such a curve,
which means d (p,q) is an honest metric. Define a Carnot-Carath´eodory ball of radius
C
r centered at a point p by
0
B(p ,r) = {p ∈ G : d (p,p ) < r}.
0 C 0
THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS 3
In addition to the Carnot-Carath´eodory metric, there is a smooth (off the origin)
gauge. This gauge is defined for a point p = (ζ ,ζ ,...,ζ ) with ζ ∈ V by
1 2 l i i
1
l 2l!
2l!
(2.2) N(p) = kζ k i
i
(cid:18) (cid:19)
Xi=1
and it induces a metric d that is bi-Lipschitz equivalent to the Carnot-Carath´eodory
N
metric and is given by
d (p,q) = N(p−1 ·q).
N
We define a gauge ball of radius r centered at a point p by
0
B (p ,r) = {p ∈ G : d (p,p ) < r}.
N 0 N 0
In this environment, a smooth function u : G → R has the horizontal derivative given
by
∇ u = (X u,X u,...,X u)
0 1 2 n1
andthesymmetrized horizontalsecondderivativematrix, denotedby(D2u)⋆, withentries
1
((D2u)⋆) = (X X u+X X u)
ij i j j i
2
for i,j = 1,2,...,n . We also consider the semi-horizontal derivative given by
1
∇ u = (X u,X u,...,X u,Y u,Y u,...,Y u).
1 1 2 n1 1 2 n2
Using the above derivatives, we define the h-homogeneous infinite Laplace operator for
h ≥ 1 by
n1
∆h f = k∇ fkh−3 X fX fX X f = k∇ fkh−3h(D2f)⋆∇ f,∇ fi.
∞ 0 i j i j 0 0 0
iX,j=1
Given T > 0 anda function u : G×[0,T] → R, we may define the analogoussubparabolic
infinite Laplace operator by
u −∆h u
t ∞
and we consider the corresponding equation
(2.3) u −∆h u = 0.
t ∞
We note that when h ≥ 3, this operator is continuous. When h = 3, we have the
subparabolic infinite Laplace equation analogous to the infinite Laplace operator in [5].
The Euclidean analog for h = 1 has been explored in [14] and the Euclidean analog for
1 < h < 3 in [15].
We recall that for any open set O ⊂ G, the function f is in the horizontal Sobolev
spaceW1,p(O)if f andX f areinLp(O) fori = 1,2,...,n . Replacing Lp(O)byLp (O),
i 1 loc
the space W1,p(O) is defined similarly. The space W1,p(O) is the closure in W1,p(O) of
loc 0
smooth functions with compact support. In addition, we recall a function u : G → R
is C2 if ∇ u and X X u are continuous for all i,j = 1,2,...n . Note that C2 is not
sub 1 i j 1 sub
equivalent to (Euclidean) C2. For spaces involving time, the space C(t ,t ;X) consists
1 2
4 THOMAS BIESKE ANDERIN MARTIN
of all continuous functions u : [t ,t ] → X with max ku(·,t)k < ∞. A similar
1 2 t1≤t≤t2 X
definition holds for Lp(t ,t ;X).
1 2
Given an open box O = (a ,b ) × (a ,b ) × ··· × (a ,b ), we define the parabolic
1 1 2 2 N N
space O to be O×[t ,t ]. Its parabolic boundary is given by ∂ O = (O×{t })∪
t1,t2 1 2 par t1,t2 1
(∂O ×(t ,t ]).
1 2
Finally, recall that if G is a Carnot group with homogeneous dimension Q, then G×R
is again a Carnot group of homogeneous dimension Q + 1 where we have added an
extra vector field ∂ to the first layer of the grading. This allows us to give mean-
∂t
ing to notations such as W1,2(O ) and C2 (O ) where we consider ∇ u to be
t1,t2 sub t1,t2 0
X u,X u,...,X u, ∂u .
1 2 n1 ∂t
(cid:0) (cid:1)
3. Parabolic Jets and Viscosity Solutions
3.1. Parabolic Jets. In this subsection, we recall the definitions of the parabolic jets,
as given in [6], but included here for completeness. We define the parabolic superjet of
u(p,t) at the point (p ,t ) ∈ O , denoted P2,+u(p ,t ), by using triples (a,η,X) ∈
0 0 t1,t2 0 0
R×V ⊕V ×Sn1 so that (a,η,X) ∈ P2,+u(p ,t ) if
1 2 0 0
\ 1
u(p,t) ≤ u(p ,t )+a(t−t )+hη,p−1·pi+ hXp−1 ·p,p−1 ·pi
0 0 0 0 2 0 0
+o(|t−t |+|p−1 ·p|2) as (p,t) → (p ,t ).
0 0 0 0
We recall that Sk is the set of k × k symmetric matrices and n = dimV . We define
i i
\
p−1 ·p as the first n coordinates of p−1 ·p and p−1 ·p as the first n +n coordinates of
0 1 0 0 1 2
p−1·p. This definition isanextension of thesuperjet definition forsubparabolic equations
0
in the Heisenberg group [4]. We define the subjet P2,−u(p ,t ) by
0 0
P2,−u(p ,t ) = −P2,+(−u)(p ,t ).
0 0 0 0
2,+
We define the set theoretic closure of the superjet, denoted P u(p ,t ), by requiring
0 0
2,+
(a,η,X) ∈ P u(p ,t ) exactly when there is a sequence
0 0
(a ,p ,t ,u(p ,t ),η ,X ) → (a,p ,t ,u(p ,t ),η,X) with the triple
n n n n n n n 0 0 0 0
(a ,η ,X ) ∈ P2,+u(p ,t ). A similar definition holds for the closure of the subjet.
n n n n n
We may also define jets using appropriate test functions. Given a function u : O →
t1,t2
R we consider the set Au(p ,t ) given by
0 0
Au(p ,t ) = {φ ∈ C2 (O ) : u(p,t)−φ(p,t) ≤ u(p ,t )−φ(p ,t ) = 0 ∀(p,t) ∈ O }.
0 0 sub t1,t2 0 0 0 0 t1,t2
consisting of all test functions that touch u from above at (p ,t ). We define the set of
0 0
all test functions that touch from below, denoted Bu(p ,t ), similarly.
0 0
The following lemma relates the test functions to jets. The proof is identical to Lemma
3.1 in [4], but uses the (smooth) gauge N(p) instead of Euclidean distance.
Lemma 3.1.
P2,+u(p ,t ) = {(φ (p ,t ),∇φ(p ,t ),(D2φ(p ,t ))⋆) : φ ∈ Au(p ,t )}.
0 0 t 0 0 0 0 0 0 0 0
THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS 5
3.2. Jet Twisting. We recall that the set V = span{X ,X ,...,X } and notationally,
1 1 2 n1
we will always denote n by n. The vectors X at the point p ∈ G can be written as
1 i
N
∂
X (p) = a (p)
i ij
∂x
Xj=1 j
forming the n×N matrix A with smooth entries A = a (p). By linear independence
ij ij
of the X , A has rank n. Similarly,
i
N
∂
Y (p) = b (p)
i ij
∂x
Xj=1 j
forming the n ×N matrix B with smooth entries B = b . The matrix B has rank n .
2 ij ij 2
The following lemma differs from [5, Corollary 3.2] only in that there is now a parabolic
term. This term however, does not need to be twisted. The proof is then identical, as
only the space terms need twisting.
Lemma 3.2. Let (a,η,X) ∈ P2,+u(p,t). (Recall that (η,X) ∈ RN ×SN.) Then
eucl
(a,A·η ⊕B·η, AXAT +M) ∈ P2,+u(p,t).
M
Here the entries of the (symmetric) matrix are given by
N N
∂ ∂a
ik
a (p) a (p)+a (p) (p) η i 6= j,
 (cid:18) il ∂x jk jl ∂x (cid:19) k
Xk=1Xl=1 l l

M = 
ij 


N N
∂a
ik
a (p) (p)η i = j.
 il ∂x k
Xk=1Xl=1 l



3.3. Viscosity Solutions. We consider parabolic equations of the form
(3.1) u +F(t,p,u,∇ u,(D2u)⋆) = 0
t 1
for continuous and proper F : [0,T] × G × R × g × Sn → R. [8] We recall that Sn is
the set of n × n symmetric matrices (where dimV = n) and the derivatives ∇ u and
1 1
(D2u)⋆ are taken in the space variable p. We then use the jets to define subsolutions and
supersolutions to Equation (3.1) in the usual way.
Definition 1. Let (p ,t ) ∈ O be as above. The upper semicontinuous function u
0 0 t1,t2
is a parabolic viscosity subsolution in O if for all (p ,t ) ∈ O we have (a,η,X) ∈
t1,t2 0 0 t1,t2
2,+
P u(p ,t ) produces
0 0
a+F(t ,p ,u(p ,t ),η,X) ≤ 0.
0 0 0 0
A lower semicontinuous function u is a parabolic viscosity supersolution in O if for all
t1,t2
2,−
(p ,t ) ∈ O we have (b,ν,Y) ∈ P u(p ,t ) produces
0 0 t1,t2 0 0
b+F(t ,p ,u(p ,t ),ν,Y) ≥ 0.
0 0 0 0
6 THOMAS BIESKE ANDERIN MARTIN
A continuous function u is a parabolic viscosity solution in O if it is both a parabolic
t1,t2
viscosity subsolution and parabolic viscosity supersolution.
Remark 3.3. In the special case when F(t,p,u,∇ u,(D2u)⋆) = Fh(∇ u,(D2u)⋆) =
1 ∞ 0
−∆h u, for h ≥ 3, we use the terms “parabolic viscosity h-infinite supersolution”, etc.
∞
In the case when 1 ≤ h < 3, the definition above is insufficient due to the singularity
occurring when the horizontal gradient vanishes. Therefore, following [14] and [15], we
define viscosity solutions to Equation (2.3) when 1 ≤ h < 3 as follows:
Definition 2. Let O be as above. A lower semicontinuous function v : O → R is
t1,t2 t1,t2
a parabolic viscosity h-infinite supersolution of u −∆h u = 0 if whenever (p ,t ) ∈ O
t ∞ 0 0 t1,t2
and φ ∈ Bu(p ,t ), we have
0 0
φ (p ,t )−∆h φ(p ,t ) ≥ 0 when ∇ φ(p ,t ) 6= 0
t 0 0 ∞ 0 0 0 0 0

 φ (p ,t )− minh(D2φ)⋆(p ,t ) η,ηi ≥ 0 when ∇ φ(p ,t ) = 0 and h = 1
 t 0 0 0 0 0 0 0
 kηk=1
φ (p ,t ) ≥ 0 when ∇ φ(p ,t ) = 0 and 1 < h < 3
t 0 0 0 0 0


An upper semicontinuous function u : O → R is a parabolic viscosity h-infinite subso-
t1,t2
lution of u −∆h u = 0 if whenever (p ,t ) ∈ O and φ ∈ Au(p ,t ), we have
t ∞ 0 0 t1,t2 0 0
φ (p ,t )−∆h φ(p ,t ) ≤ 0 when ∇ φ(p ,t ) 6= 0
t 0 0 ∞ 0 0 0 0 0

 φ (p ,t )− maxh(D2φ)⋆(p ,t ) η,ηi ≤ 0 when ∇ φ(p ,t ) = 0 and h = 1
 t 0 0 0 0 0 0 0
 kηk=1
φ (p ,t ) ≤ 0 when ∇ φ(p ,t ) = 0 and 1 < h < 3
t 0 0 0 0 0



Acontinuous function is a parabolic viscosity h-infinite solution if it is both a parabolic
viscosity h-infinite subsolution and parabolic viscosity h-infinite subsolution.
Remark 3.4. When 1 < h < 3, we can actually consider the continuous operator
−k∇ ukh−3h(D2u)⋆∇ u,∇ ui = −∆h u ∇ u 6= 0
(3.2)Fh(∇ u,(D2u)⋆) = 0 0 0 ∞ 0
∞ 0 (cid:26) 0 ∇ u = 0.
0
Definitions 1 and 2 would then agree. (cf. [15])
We also wish to define what [12] refers to as parabolic viscosity solutions. We first
need to consider the set
A−u(p ,t ) = {φ ∈ C2(O ) : u(p,t)−φ(p,t) ≤ u(p ,t )−φ(p ,t ) = 0forp 6= p ,t < t }
0 0 t1,t2 0 0 0 0 0 0
consisting of all functions that touch from above only when t < t . Note that this set is
0
larger than Au and corresponds physically to the past alone playing a role in determining
the present. We define B−u(p ,t ) similarly. We then have the following definition.
0 0
Definition 3. An upper semicontinuous function u on O is a past parabolic viscosity
t1,t2
subsolution in O if φ ∈ A−u(p ,t ) produces
t1,t2 0 0
φ (p ,t )+F(t ,p ,u(p ,t ),∇ φ(p ,t ),(D2φ(p ,t ))⋆) ≤ 0.
t 0 0 0 0 0 0 1 0 0 0 0
THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS 7
An lower semicontinuous function u on O is a past parabolic viscosity supersolution
t1,t2
in O if φ ∈ B−u(p ,t ) produces
t1,t2 0 0
φ (p ,t )+F(t ,p ,u(p ,t ),∇ φ(p ,t ),(D2φ(p ,t ))⋆) ≥ 0.
t 0 0 0 0 0 0 1 0 0 0 0
A continuous function is a past parabolic viscosity solution if it is both a past parabolic
viscosity supersolution and subsolution.
We have the following proposition whose proof is obvious.
Proposition 3.5. Past parabolic viscosity sub(super-)solutions are parabolic viscosity
sub(super-)solutions. In particular, past parabolic viscosity h-infinite sub(super-)solutions
are parabolic viscosity h-infinite subsub(super-)solutions for h ≥ 1.
3.4. The Carnot Parabolic Maximum Principle. In this subsection, we recall the
Carnot Parabolic Maximum Principle and key corollaries, as proved in [6].
Lemma 3.6 (Carnot Parabolic Maximum Principle). Let u be a viscosity subsolution
to Equation (3.1) and v be a viscosity supersolution to Equation (3.1) in the bounded
parabolic set Ω × (0,T) where Ω is a (bounded) domain and let τ be a positive real
parameter. Let φ(p,q,t) = ϕ(p · q−1,t) be a C2 function in the space variables p and q
and a C1 function in t. Suppose the local maximum
(3.3) M ≡ max {u(p,t)−v(q,t)−τφ(p,q,t)}
τ
Ω×Ω×[0,T]
occurs at the interior point (p ,q ,t ) of the parabolic set Ω×Ω×(0,T). Define the n×n
τ τ τ
matrix W by
W = X (p)X (q)φ(p ,q ,t ).
ij i j τ τ τ
Let the 2n×2n matrix W be given by
0 1(W −WT)
(3.4) W = 2
(cid:18)1(WT −W) 0 (cid:19)
2
and let the matrix W ∈ S2N be given by
D2 φ(p ,q ,t ) D2 φ(p ,q ,t )
pp τ τ τ pq τ τ τ
(3.5) W =  
D2 φ(p ,q ,t ) D2 φ(p ,q ,t )
 qp τ τ τ qq τ τ τ .
Suppose
lim τφ(p ,q ,t ) = 0.
τ τ τ
τ→∞
Then for each τ > 0, there exists real numbers a and a , symmetric matrices X and Y
1 2 τ τ
and vector Υ ∈ V ⊕V , namely Υ = ∇ (p)φ(p ,q ,t ), so that the following hold:
τ 1 2 τ 1 τ τ τ
2,+ 2,−
A) (a ,τΥ ,X ) ∈ P u(p ,t ) and (a ,τΥ ,Y ) ∈ P v(q ,t ).
1 τ τ τ τ 2 τ τ τ τ
B) a −a = φ (p ,q ,t ).
1 2 t τ τ τ
8 THOMAS BIESKE ANDERIN MARTIN
C) For any vectors ξ,ǫ ∈ V , we have
1
hX ξ,ξi−hY ǫ,ǫi ≤ τh(D2φ)⋆(p ,q ,t )(ξ −ǫ),(ξ −ǫ)i+τhW(ξ ⊕ǫ),(ξ ⊕ǫ)i
τ τ p τ τ τ
(3.6) +τkWk2kA(pˆ)Tξ ⊕A(qˆ)Tǫk2.
In particular,
(3.7) hX ξ,ξi−hY ξ,ξi . τkWk2kξk2.
τ τ
Corollary 3.7. Let φ(p,q,t) = φ(p,q) = ϕ(p·q−1) be independent of t and a non-negative
function. Suppose φ(p,q) = 0 exactly when p = q. Then
lim τφ(p ,q ) = 0.
τ τ
τ→∞
In particular, if
N
1
(3.8) φ(p,q,t) = (p·q−1) m
i
m
Xi=1 (cid:0) (cid:1)
for some even integer m ≥ 4 where (p·q−1) is the i-th component of the Carnot group
i
multiplication group law, then for the vector Υ and matrices X ,Y , from the Lemma,
τ τ τ
we have
2,+ 2,−
A) (a ,τΥ ,X ) ∈ P u(p ,t ) and (a ,τΥ ,Y ) ∈ P v(q ,t ).
1 τ τ τ τ 1 τ τ τ τ
B) The vector Υ satisfies
τ
m−1
kΥτk ∼ φ(pτ,qτ) m .
C) For any fixed vector ξ ∈ V , we have
1
(3.9) hXτξ,ξi−hYτξ,ξi . τkWk2kξk2 . τ(φ(pτ,qτ))2mm−4kξk2.
4. Uniqueness of viscosity solutions
We wish to formulate a comparison principle for the following problem.
Problem 4.1. Let h ≥ 1. Let Ω be a bounded domain and let Ω = Ω × [0,T). Let
T
ψ ∈ C(Ω) and g ∈ C(Ω ). We consider the following boundary and initial value problem:
T
u +Fh(∇ u,(D2u)⋆) = 0 in Ω×(0,T) (E)
t ∞ 0
(4.1)  u(p,t) = g(p,t) p ∈ ∂Ω, t ∈ [0,T) (BC)
 u(p,0) = ψ(p) p ∈ Ω (IC)
We also adopt the definition that a subsolution u(p,t) to Problem 4.1 is a viscosity sub-
solution to (E), u(p,t) ≤ g(p,t) on ∂Ω with 0 ≤ t < T and u(p,0) ≤ ψ(p) on Ω.
Supersolutions and solutions are defined in an analogous matter.
Because our solution u will be continuous, we offer the following remark:
Remark 4.2. The functions ψ and g may be replaced by one function g ∈ C(Ω ). This
T
combines conditions (E) and (BC) into one condition
(4.2) u(p,t) = g(p,t), (p,t) ∈ ∂ Ω (IBC)
par T
THE PARABOLIC INFINITE-LAPLACE EQUATION IN CARNOT GROUPS 9
Theorem 4.3. Let Ω be a bounded domain in G and let h ≥ 1. If u is a parabolic
viscosity subsolution and v a parabolic viscosity supersolution to Problem (4.1) then u ≤ v
on Ω ≡ Ω×[0,T).
T
Proof. Our proof follows that of [8, Thm. 8.2] and so we discuss only the main parts.
ε
For ε > 0, we substitute u˜ = u− for u and prove the theorem for
T−t
ε
(4.3) u +Fh(∇ u,(D2u)⋆) ≤ − < 0
t ∞ 0 T2
(4.4) limu(p,t) = −∞ uniformly on Ω
t↑T
and take limits to obtain the desired result. Assume the maximum occurs at (p ,t ) ∈
0 0
Ω×(0,T) with
u(p ,t )−v(p ,t ) = δ > 0.
0 0 0 0
Case 1: h > 1.
Let H ≥ h+3 be an even number. As in Equation (3.8), we let
N
1
φ(p,q) = (p·q−1) H
i
H
Xi=1 (cid:0) (cid:1)
where (p·q−1) is the i-th component of the Carnot group multiplication group law. Let
i
M = u(p ,t )−v(q ,t )−τφ(p ,q )
τ τ τ τ τ τ τ
with (p ,q ,t ) the maximum point in Ω×Ω×[0,T) of u(p,t)−v(q,t)−τφ(p,q).
τ τ τ
If t = 0, we have
τ
0 < δ ≤ M ≤ sup(ψ(p)−ψ(q)−τφ(p,q))
τ
Ω×Ω
leading to a contradiction for large τ. We therefore conclude t > 0 for large τ. Since
τ
u ≤ v on ∂Ω × [0,T) by Equation (BC) of Problem (4.1), we conclude that for large
τ, we have (p ,q ,t ) is an interior point. That is, (p ,q ,t ) ∈ Ω × Ω× (0,T). Using
τ τ τ τ τ τ
Corollary 3.7 Property A, we obtain
2,+
(a,τΥ(p ,q ),X ) ∈ P u(p ,t )
τ τ τ τ τ
2,−
(a,τΥ(p ,q ),Y ) ∈ P v(q ,t )
τ τ τ τ τ
satisfying the equations
ε
a+Fh(τΥ(p ,q ),X ) ≤ −
∞ τ τ τ T2
a+Fh(τΥ(p ,q ),Y ) ≥ 0.
∞ τ τ τ
10 THOMAS BIESKE ANDERIN MARTIN
Ifthereisasubsequence {p ,q } suchthatp 6= q , wesubtract, andusingCorollary
τ τ τ>0 τ τ
3.7, we have
ε
0 < ≤ (τΥ(p ,q ))h−3τ2 hX Υ(p ,q ),Υ(p ,q )i−hY Υ(p ,q ),Υ(p ,q )i
τ τ τ τ τ τ τ τ τ τ τ τ
T2 (cid:18) (cid:19)
(4.5) . τh ϕ(pτ,qτ)HH−1 h−3(ϕ(pτ,qτ))2HH−4(ϕ(pτ,qτ))2HH−2
(4.6) = τh((cid:0)ϕ(pτ,qτ))Hh+(cid:1)HH−h−3 = (τϕ(pτ,qτ))hϕ(pτ,qτ)H−Hh−3.
Because H > h+3, we arrive at a contradiction as τ → ∞.
If we have p = q , we arrive at a contradiction since
τ τ
Fh(τΥ(p ,q ),X ) = Fh(τΥ(p ,q ),Y ) = 0.
∞ τ τ τ ∞ τ τ τ
Case 2: h = 1.
We follow the proof of Theorem 3.1 in [14]. We let
N
1 1
ϕ(p,q,t,s) = (p·q−1) 4 + (t−s)2
i
4 2
Xi=1 (cid:0) (cid:1)
and let (p ,q ,t ,s ) be the maximum of
τ τ τ τ
u(p,t)−v(q,s)−τφ(p,q,t,s)
Again, for large τ, this point is an interior point. If we have a sequence where p 6= q ,
τ τ
then Lemma 3.2 yields
2,+
(τ(t −s ),τΥ(p ,q ),X ) ∈ P u(p ,t )
τ τ τ τ τ τ τ
2,−
(τ(t −s ),τΥ(p ,q ),Y ) ∈ P v(q ,s )
τ τ τ τ τ τ τ
satisfying the equations
ε
τ(t −s )+Fh(τΥ(p ,q ),X ) ≤ −
τ τ ∞ τ τ τ T2
τ(t −s )+Fh(τΥ(p ,q ),Y ) ≥ 0.
τ τ ∞ τ τ τ
As in the first case, we subtract to obtain
ε
0 < ≤ (τΥ(p ,q ))−2τ2 hX Υ(p ,q ),Υ(p ,q )i−hY Υ(p ,q ),Υ(p ,q )i
τ τ τ τ τ τ τ τ τ τ τ τ
T2 (cid:18) (cid:19)
. ϕ(pτ,qτ)−23(τϕ(pτ,qτ)ϕ(pτ,qτ)32) = τϕ(pτ,qτ).
We arrive at a contradiction as τ → ∞.
If p = q , then v(q,s)−βv(q,s) has a local minimum at (q ,s ) where
τ τ τ τ
N
τ τ
βv(q,s) = − (p ·q−1) 4 − (t −s)2.
τ i τ
4 2
Xi=1 (cid:0) (cid:1)
We then have
0 < ε(T −s )−2 ≤ βv(q ,s )− minh(D2βv)⋆(q ,s ) η,ηi.
τ s τ τ τ τ
kηk=1