On the nature of isolated asymptotic singularities of solutions of a family of quasi-linear elliptic PDE's on a Cartan-Hadamard manifold

0.15 MB·

by  Leonardo Prange Bonorino

On the nature of isolated asymptotic singularities of
6
1
0 solutions of a family of quasi-linear elliptic PDE’s
2
on a Cartan-Hadamard manifold
r
p
A
Leonardo Bonorino Jaime Ripoll
9
2

]
Abstract
G
D Let M be a Cartan-Hadamard manifold with sectional curvature
. satisfying b2 K a2 < 0, b a > 0. Denote by ∂∞M the
h asymptotic−bou≤ndary≤of−M and by M≥¯ := M ∂ M the geometric
t ∪ ∞
a compactificationofM withtheconetopology. Weinvestigateherethe
m
following question: Given a finite number of points p1,...,pk ∂∞M,
[ if u C∞(M) C0 M¯ p1,...,pk satisfies a PDE (u) =∈0 in M
∈ ∩ \{ } Q
and if u extends continuously to p , i=1,...,k, canone
v2 conclude|∂∞thMat\{up1,...,Cpk(cid:0)0} M¯ ? When d(cid:1)imM = 2, ifor belonging to a
∈ Q
1 linearly convex space of quasi-linear elliptic operators of the form
6 (cid:0) (cid:1) S
3 ( u)
(u)=div A |∇ | u =0,
0 Q u ∇
0 (cid:18) |∇ | (cid:19)
1. where satisfies some structural conditions, then the answer is yes
A
0 provided that has a certain asymptotic growth. This condition in-
A
6 cludes,besidestheminimalgraphPDE,aclassofminimaltypePDEs.
1 In the hyperbolic space Hn, n 2, we are able to give a complete
: ≥
v answer: we prove that splits into two disjoint classes of minimal
S
i type and p Laplaciantype PDEs, p>1, where the answer is yes and
X −
no respectively. These two classes are determined by the asymptotic
r
a behaviour of . Regarding the class where the answer is negative, we
A
obtain explicit solutions having an isolated non removable singularity
at infinity.
1 Introduction
Let M be Cartan-Hadamard n dimensional manifold (complete, con-
−
nected, simply connected Riemannian manifold with non-positive sectional
curvature). It is well-known that M can be compactified with the so called
cone topology by adding a sphere at infinity, also called the asymptotic
1
boundary of M; we refer to [4] for details. In the sequel, we will denote by
∂ M the sphere at infinity and by M¯ = M ∂ M the compactification of
∞ ∞
∪
M.
We recall that the asymptotic Dirichlet problem of a PDE (u) = 0 in
Q
M for a given asymptotic boundary data ψ C0(∂ M) consists in finding
∞
∈
a solution u C0 M¯ of Q(u) = 0 in M such that u = ψ, determining
∈ |∂∞M
the uniqueness of u as well.
(cid:0) (cid:1)
Theasymptotic Dirichlet problem for the Laplacian PDE has been stud-
ied during the last 30 years and there is a vast literature in this case. More
recently, it has been studied in a larger class of PDEs which include the
p Laplacian PDE, p > 1,
−
u
∆ u= div ∇ = 0,
p up
|∇ |
see [7], and the minimal graph PDE,
u
(u) = div ∇ = 0, (1)
M 1+ u2
|∇ |
q
see [6], [10], case that we are specially interested in the present work. We
note that div and are the divergence and the gradient in M and it is
∇
worth to mention that the graph
G(r) = (x,u(x)) x M
{ | ∈ }
of u is a minimal surface in M R if and only if u satisfies (1).
×
PresentlyitisknownthattheasymptoticDiricheltproblemcanbesolved
in any Cartan-Hadamard manifold under hypothesis on the growth of the
sectionalcurvaturethatincludestheoneswithnegativelypinchedcurvature,
for any given continuous data at infinity, and on a large class of PDEs that
includes both p Laplacian and minimal graph PDEs (see [2], [11]).
−
A natural question related to the asymptotic Dirichlet problem concerns
the existence or not of solutions with isolated singularities at ∂ M. We
∞
investigate this problem on the following class of quasi-linear elliptic op-
S
erators:
( u)
(u) = div A |∇ | u = 0, (2)
Q u ∇
(cid:18) |∇ | (cid:19)
2
where C1[0, ) satisfies the following conditions:
A ∈ ∞
(0) = 0, ′(s) > 0for s > 0;
A A
(s) C(sp−1+1) for some C >0, somep 1 and any s >0; (3)

A ≤ ≥
there exist positives q, δ0 andD¯ s.t. (s) >D¯sq for s [0,δ0]. 
A ∈

This class of operators, as the authors know, was first introduced and
studied regardingthesolvability of theasymptotic Dirichlet problem in[11];
it includes well known geometric operators as the p-laplacian, for p > 1,
( (s) = sp−1) and the minimal graph operator ( (s) = s/√1+s2). Note
A A
that is linearly convex that is, any two elements , of are homoth-
1 2
S Q Q S
opic in by the line segment t +(1 t) , 0 t 1.
1 2
S Q − Q ≤ ≤
As we shall see, the nature of an isolated asymptotic singularity of
Q
depends on the asymptotic behaviour of and can change drastically ac-
A
cordingly to it. It is worth to mention at this point that this behaviour of
is closely related to the existence or not of “Scherk type” solutions of
A
(2) (see the beginning of the next section). Minimal Scherk surfaces play
a fundamental role on the theory of minimal surfaces in Riemannian mani-
folds (a well known breakthrough result using Scherk minimal surfaces were
obtained by P. Collin and H. Rosenberg in [3]).
In our first three results we are concerned with removable singularities.
We first show that isolated singularities are removable if n = 2, M has
negatively pinched curvature and satisfies
A
∞
−1(K (cosh(ar))−1)dr = + ,
0
A ∞
Z0
for some K > 0. Since −1(t) ct1/q holds for small t, due to (3), the
0
A ≤
change of variable t = K (cosh(ar))−1 implies that this condition is equiva-
0
lent to
K0 −1(t)
A dt = + . (4)
√K t ∞
Z0 0−
Precisely, we prove:
Theorem1.1. Suppose that M isa2 dimensional Cartan-Hadamard man-
−
ifold with sectional curvature satisfying b2 K a2 < 0, b a >
− ≤ ≤ − ≥
0. Given a finite number of points p ,...,p ∂ M, if m C∞(M)
1 k ∞
∈ ∈ ∩
C0 M¯ p ,...,p is a solution of (2) in M, (s) satisfies (3) and (4),
1 k
\{ } A
and m extends continuously to p , i = 1,...,k, then m
(cid:0) |∂∞M\{p1,...,p(cid:1)k} i ∈
C0 M¯ .
(cid:0) (cid:1)
3
We observe that condition (4) fails if K < sup . Hence, (4) implies
0
A
that is bounded and K = sup . This happens, for instance, if (s) =
0
A A A
s/√1+s2. Therefore, we have
Corollary1.2. Suppose thatM isa2 dimensionalCartan-Hadamard man-
−
ifold with sectional curvature satisfying b2 K a2 < 0, b a >
− ≤ ≤ − ≥
0. Given a finite number of points p ,...,p ∂ M, if m C∞(M)
1 k ∞
∈ ∈ ∩
C0 M¯ p ,...,p is a solution of the minimal surface equation and if
1 k
\{ }
m extends continuously to p , i = 1,...,k, then m C0 M¯ .
|∂(cid:0)∞M\{p1,...,pk} (cid:1) i ∈
We observe that a similar problem can obviously be posed to sol(cid:0)utio(cid:1)ns
of (2) on a bounded C0 domain Ω of R2. In the minimal case, this a an old
problem. From a classical result of R. Finn [5], it follows that if u, as in the
above theorem, with M replaced by Ω,∂ by ∂, is a solution of the minimal
∞
graph equation (1) and if there there is a solution v C∞(Ω) C0 Ω¯ of
∈ ∩
(1) such that
(cid:0) (cid:1)
u = v
|∂Ω\{p1,...,pn} |∂Ω\{p1,...,pn}
then u = v and hence u extends continuously through the singularities.
If the Dirichlet problem (u) = 0 on Ω is not solvable for the continuous
M
boundarydataφ:= u thentheresultisfalse,aknownfactontheclassical
∂Ω
|
minimalsurfacetheory(see[9], ChapterV,Section3). Weremarkthateven
if the Dirichlet problem is not solvable there might exist smooth compact
minimal surfaces which boundary is the graph of φ if φ and the domain are
regular enough (see [1]).
Although under the hypothesis of Corollary 1.2 there exists a solution
v C∞(M) C0 M¯ of (1) such that u = v ,
∈ ∩ |∂∞M\{p1,...,pn} |∂∞M\{p1,...,pn}
we felt necessary to use a different approach from Finn’s. First because
(cid:0) (cid:1)
the boundedness of the domain is fundamental to the arguments used in
[5]. Secondly, because it is not clear that the asymptotic Dirichlet problem
for the PDE (2), under the conditions (3), is solvable for any continuous
boundary data given at infinity.
Ourproofreliesheavilyonasymptoticpropertiesof2 dimensionalCartan-
−
Hadamard manifolds. It is fundamentally based on the fact that a point p
of the asymptotic boundary of M is an isolated point of the asymptotic
boundary of a domain U such that M U is convex. This property allows
\
the construction of suitable barriers at infinity. Although the existence of U
inthen = 2dimensionalcaseistrivial (forexample, adomainwhichbound-
ary are two geodesics asymptotic to p), we don’t know if such an U exists in
M if n 3. Nevertheless, it is possible in the special case of the hyperbolic
≥
space to give an ad hoc proof of Theorem 1.1 using the symmetries of the
space. Precisely, our result in Hn reads:
4
Theorem 1.3. Let Hn be the hyperbolic space of constant section curvature
1. Given a finite number of points p ,...,p ∂ Hn, if m C∞(Hn)
1 k ∞
−C0 H¯n p ,...,p is a solution of (2) in Hn∈, (s) satisfies∈(3) and (4)∩,
1 k
\{ } A
Can0d(cid:0)Hi¯fnm.|∂∞Hn\{p1,..(cid:1).,pk} extends continuously to pi, i = 1,...,k, then m ∈
(cid:0)Fina(cid:1)lly, in the next last result, we prove the existence of a class of solu-
tions of (2) in Hn admiting a non removable isolated asymptotic singularity.
Note that this class contains the p Laplacian PDE, p > 1.
−
Theorem 1.4. Suppose that (3) holds and (s) is unbounded. Given a
point p ∂ Hn, there exists a solution m AC∞(Hn) C0 H¯n p of
1 ∞ 1
∈ ∈ ∩ \{ }
(2) in Hn, such that m = 0 on ∂ Hn p and limsup m = + .
∞ \{ 1} x→p1 (cid:0) ∞ (cid:1)
2 Proof of the theorems
We begin by constructing Scherk type supersolutions to the equation (2),
which are fundamental to prove the nonexistence of true asymptotic singu-
larities.
Lemma 2.1. Let γ be some geodesic of M, let U be one of the connected
component of M γ and δ > 0. If satisfies (3) and (4), then there exists
\ A
a solution of
( u)
div A |∇ | u 0 in U
u ∇ ≤
 (cid:18) |∇ | (cid:19)
u = + on γ

 ∞

u = δ in int ∂ U.
∞




Proof. Let d : U R be defined by d(x) = dist(x,γ) and g : (0,+ ) R
→ ∞ →
be defined by
∞ K
g(d) = δ+ −1 0 dt,
A cosh(at)
Zd (cid:18) (cid:19)
where K = sup . Observe that according to [11], g(d) is well defined and
0
A
finite for all d > 0, and v(x) := g(d(x)) is a supersolution of (2). Moreover,
g(d) δ as d + and, therefore, g(d(x)) δ as x p ∂ U.
∞
→ → ∞ → → ∈
That is, v = δ on int ∂ U. Finally, making the change of variable z =
∞
K (cosh(at))−1, we can prove that condition (4) implies that g(d) +
0
→ ∞
as d 0. Hence v(x) = g(d(x)) + as x x γ, completing the
0
→ → ∞ → ∈
lemma.
5
2.1 Proof of Theorem 1.1
We first claim that m is bounded: For each p , consider a geodesic Γ
i i
such that the asymptotic boundary of one of the connected components of
M Γ ,sayX ,doesnotcontainp forj = i. Assumealsothatp int∂ X .
i i j i ∞ i
\ 6 ∈
Since Γ ( ) p ,... p , m is continuous at Γ ( ) and therefore it is
i 1 n i
±∞ 6∈ { } ±∞
bounded on Γ . Let S = supm for i 1,...n , S = supm
i i ∈ { } 0 |∂∞M\{p1,...,pn}
Γi
and
S = max S ,S ,...,S .
0 1 n
{ }
From the maximum principle, m S in M X X . To prove that
1 n
≤ \{ ∪···∪ }
m S in X , take a sequence of geodesics β such that the ending points
i k
≤
β (+ ) and β ( ) converge to p . Let Y be the connected component
k k i k
∞ −∞
of M β whose the asymptotic boundary does not contain p . Observe that
k i
\
M X Y for large k and Y = M. Let w be the supersolution of (2)
i k k k
\ ⊂ ∪
givenbyLemma2.1. Recallthatw is+ onβ andS at∂ Y β ( ) .
k k ∞ k k
∞ \{ ±∞ }
Hence w S and therefore w m on Γ = ∂X , w = S m on
k k i i k
≥ ≥ ≥
∂ (X Y ) and w = + > m on β = ∂Y . Then w m in Y X for
∞ i k k k k k k i
∩ ∞ ≥ ∩
large k by the Comparison Principle. For any given x M, x Y for large
k
∈ ∈
k. Hence, using that w (x) S, we have m(x) S. In a similar way, we
k
→ ≤
can conclude that m is bounded from below, proving the claim.
Assume that m S. Denote by φ the continuous extension of
≤
m to ∂ M. Let p p ,...,p . Adding a constant to φ we
|∂∞M\{p1,...,pn} ∞ ∈ { 1 n}
may assume wlg that φ(p)= 0. Let 0 < δ S be given. We will prove that
≤
K := limsup m(x) δ. By contradiction assume that that K > δ.
x→p
≤
By the continuity of φ, there exists an open connected neighborhood
∂ M of p such that φ(q) δ for all q . Moreover, we may assume
∞
O ⊂ ≤ ∈ O
that does not contain another point p except p.
i
O
Let γ be a geodesic such that γ( ) = p. Set γ = γ(R). Choose a point
∞
q γ and a geodesic α orthogonal to γ at q such that α ( ) . Let
0 0 0 0
∈ ±∞ ∈ O
γ , i 1,2 , be the geodesics with ending points at p and q := α ( ) and
i 1 0
∈ { } ∞
p and q := α ( ), respectively. Denote by U the connected component
2 0 i
−∞
of M γ that does not contain α . As before, there exists Sh solution of
i 0 i
\
( u)
div A |∇ | u 0 in U
i
u ∇ ≤
 (cid:18) |∇ | (cid:19)
u = + on γ
 ∞ i

u = δ in int ∂ U .
∞ i




6
Observe that m < Sh . Let c be the level set of Sh
i i i
K δ
c = x M : Sh (x)= +
i i
∈ 2 2
(cid:26) (cid:27)
and
K δ
V = x U : Sh (x) < +
i i i
∈ 2 2
(cid:26) (cid:27)
Hence m < K/2+δ/2 on V . Let V = A (V V ).
i 1 2
\ ∪
Now, let W be a neighborhood of p (a ball centered at p) such that the
asymptotic boundaryofW V is p . ObservethatforR > 0andanypoint
∩ { }
z ontheboundaryof W V thereexistaballof radiusR,B M (W V)
R
∩ ⊂ \ ∩
such that B W V = z . We consider R =1.
R
∩ ∩ { }
Since p is an ending point of both γ and γ , the distance between any
1 2
point of W V and the geodesic γ is bounded by some constant. This
i
∩
property still holds if we consider the curve c instead γ , since these two
i i
curves are equidistant. Then there is ρ > 0 be such that
dist(x,V ) < ρ for any x W V.
i
∈ ∩
That is, for any x W V, there is a ball B centered at some point of
ρ
∈ ∩
∂(V V ) W s.t. x B .
1 2 ρ
∪ ∩ ∈
q
1
c
V 1
1
c
1
W
p V p x
B
ρ
c
2
V
2 c
2
q
2
Fig. 1 Fig. 2
7
Lemma 2.2. There exist h and h depending only on b, ρ, K and δ,
0 1
satisfying
δ
δ < h < h < K/2+
1 0
2
suchthat, foranyy M,theDirichletproblemintheannulusB (y) B (y)
2ρ+1 1
∈ \
( u)
div A |∇ | u = 0 in B (y) B (y)
2ρ+1 1
u ∇ \
 (cid:18) |∇ | (cid:19)
u = δ on ∂B (y)
 1


u = h on ∂B (y)
0 2ρ+1
has a supersolution w (x) and w (x) h if dist(x,y) < ρ+1.
 y y ≤ 1
Proof. Let f :[1, ) R be the function defined by
∞ →
r sinhbα
f(r)= δ+ −1 ds,
A sinh(bs)
Z1 (cid:18) (cid:19)
where 0 < α 1. Hence f(1) = δ and, choosing α sufficiently small,
≤
f(2ρ+1) < K/2+δ/2. Let h = f(2ρ+1). Observe that if r = r(x˜) is
0
the distance in H2( b2) from x˜ to a fixed point, then the the graphic of f
−
is a radially symmetric surface, solution of (2) in the hyperbolic plane with
constant negative sectional curvature b2, that is, f satisfies
−
′(f′(r))f′′(r)+ (f′(r))bcothbr = 0.
A A
Moreover, from the Comparison Laplacian Theorem
∆d(x) ∆r(x˜) = bcothbr,
≤
where d(x) = dist(x,y) and x˜ H2( b2) is a point such that d(x) = r(x˜).
∈ −
Then, using these two relations and that f′ > 0, we conclude that w (x) :=
y
f(d(x)) is a supersolution of (2) in M.
Since f(1) = δ and f(2ρ+1)= h , w (x) satisfies the required boundary
0 y
conditions. Finally definingh := f(ρ+1),w (x) h < h inB (y).
1 y 1 0 ρ+1
≤
Let ε be a positive real satisfying h h (K δ)/2 ε < h h and
0 1 0 1
− − − ≤ −
W W be a neighborhood of p (a ball centered at p) s.t.
0
⊂
m < K +ε in W .
0
Let W˜ W be a neighborhood of p (a ball centered at p) s.t.
0
⊂
dist(∂W ,W˜ ) > 3ρ+2.
0
8
c
1
p W˜ W
W
0
c
2
Fig. 3
We claim that
m < K +ε h +h < K
0 1
−
in W˜ .
Indeed: Let x W˜ and assume first that x V. As observed above,
∈ ∈
there is some z ∂(V V ), say z ∂V , s.t.
1 2 1
∈ ∪ ∈
x B (z)
ρ
∈
and there is y V s.t.
1
∈
B (y) W V = z .
1
∩ ∩ { }
Therefore
dist(x,y) < ρ+1.
Using triangular inequality and that dist(∂W ,W˜ ) >3ρ+2, we have
0
B (y) B (x) W .
2ρ+1 3ρ+2 0
⊂ ⊂
Let w be the solution associated to the annulus B (y) B (y) given by
y 2ρ+1 1
\
Lemma 2.2. Define
w = w +K +ε h
y 0
−
Then, using that B (y) V ,
1 1
⊂
K δ K δ
w = δ+K +ε h > K +δ+ε > + > m on ∂B (y)
0 1
− − 2 − 2 2 2
9
and, from B (y) W ,
2ρ+1 0
⊂
w = h +K +ε h = K +ε > m on ∂B (y).
0 0 2ρ+1
−
From the comparison principle,
m < w in B (y) B (y)
2ρ+1 1
\
and, therefore
m < w +K +ε h < h +K +ε h in B (y) B (y).
y 0 1 0 ρ+1 1
− − \
Since dist(x,y) < ρ+1, then x B (y). Hence, using that x V V ,
ρ+1 1 2
∈ 6∈ ∪
we have x B (y) B (y). In this case, m(x) < h +K+ε h . Finally, if
ρ+1 1 1 0
∈ \ −
x V V ,thedefinitionofεimpliesthatm(x) < K/2+δ/2 K+ε h +h
1 2 0 1
∈ ∪ ≤ −
proving the claim.
To conclude the proof of the theorem, note that ν := ε+h h > 0,
0 1
− −
since ε < h h . Then
0 1
−
K +ε h +h = K ν
0 1
− −
and, from the above claim,
m < K ν < K in W˜ .
−
Hence
limsupm(x) K ν < K
≤ −
x→p
leading a contradiction.
2.2 Proof of Theorem 1.3.
Proof. The proof that m is bounded follows the same idea as in Theorem
1.1 replacing the geodesics Γ and β by totally geodesic hyperspheres H
i k i
and Λ respectively and considering the same S. To build a supersolution
k
w such that w = + on Λ , we use the same construction as in Lemma
k k k
∞
2.1, that is, we consider
∞ K
g(d) = S + −1 0 dt,
A (cosh(at))n−1
Zd (cid:18) (cid:19)
that is well defined and finite for all d > 0. The function w (x) := g(d(x)),
k
where d(x) = dist(x,Λ ), is a supersolution according to [11]. Moreover it
k
10