Electronic Journal of Differential Equations
Vol. 1995(1995), No. 07, pp. 1-10. Published June 15, 1995.
ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp (login: ftp) 147.26.103.110 or 129.120.3.113
A Remark on ∞-harmonic Functions on
Riemannian Manifolds ∗
Nobumitsu Nakauchi
Abstract
In this note we prove an equality for ∞-harmonic functions on Riemannian manifolds. As a corollary, there is no non-constant ∞-harmonic
function on positively (or negatively) curved manifolds.
1
Introduction
In [1], [2], Aronsson studied solutions of the boundary value problem for the
degenerate elliptic equation
X
∇i u ∇j u ∇i ∇j u = 0
(1)
i,j
in a bounded subdomain D of Rn with the boundary condition u = ϕ on
∂D . His motivation is to consider the absolutely minimizing Lipschitz extension
problem, which means the problem of finding an extension u in W1,∞ (D) of
any given Lipschitz function ϕ on ∂D satisfying the minimization property
k∇ukL∞ (U) ≤ k∇vkL∞ (U)
for any open set U ⊂ D and for v ∈ W1,∞ (U ) such that v − u ∈ W01,∞ (U ) .
The equation (1) is the Euler-Lagrange equation of the functional F∞ (u) =
k∇ukL∞ in the following sense. A p-harmonic function u is a solution of
div(k∇ukp − 2 ∇u) = 0 ,
(2)
which is the Euler-Lagrange equation of the functional Fp (u) = k∇ukLp .
Rewrite (2) to read
X
1
k∇uk2 4 u +
∇i u ∇j u ∇i ∇j u = 0 .
p−2
i,j
∗ 1991 Mathematics Subject Classifications: 35J70, 35J20, 26B35.
Key words and phrases: ∞-harmonic function, ∞-Laplacian.
c
1995
Southwest Texas State University and University of North Texas.
Submitted: February 20,1995.
1
2
∞-harmonic Functions
EJDE–1995/07
Formally passing to the limit as p tends to infinity, the Euler-Lagrange equation (2) of the functional Fp converges in some sense to the Euler-Lagrange
equation (1) of the functional F∞ . From the point of view by Aronsson,
Jensen [6] obtained existence and uniqueness results. (See also Bhattacharya,
DiBenedetto and Manfredi [4].) He proved
1. any solution of the absolutely minimizing Lipschitz extension problem is
a viscosity solution of (1), and
2. there exists a unique viscosity solution of (1). Any bounded such solution
is locally Lipschitz continuous.
Aronsson’s pioneering papers [1], [2] investigated classical solutions. Recently
Evans [5] obtained a Harnack inequality for classical solutions.
The absolutely minimizing Lipschitz extension problem is considered also
on subdomains of Riemannian manifolds M . Then the associated equation
corresponding to (1) is
g ip g jq ∇i u ∇j u ∇p ∇q u = 0 ,
(3)
where gij (resp. g ij ) is the metric of M (resp. the inverse matrix of gij ),
and ∇ denotes the Levi-Civita connection of g . (Throughout this note, we use
the Einstein summation convention; if the same index appears twice, once as a
superscript and once as a subscript, then the index is summed over all possible
2,2+ε
values.) In this note we are concerned with Wloc
-solutions of (3) ( ε > 0 ).
2,2+ε
We say that u is a Wloc -solution of (3) in D if the following two conditions
hold:
1. u is locally Lipschitz continuous, and
2,2+ε
2. u ∈ Wloc
(D), and u satisfies (3) a.e.,
2,2+ε
where Wloc
(D) denotes the Sobolev space of functions whose second derivatives belong to L2+ε
loc (D) . On this general setting, the curvature of M provides
2,2+ε
an obstruction on existence of nontrivial Wloc
-solutions of (3) . The purpose
of this note is to prove the following equality.
Theorem 1 Let M be a Riemannian manifold, and let D be a domain in M .
2,2+ε
Let u be a Wloc
-solution of the equation (3) in D . Then
g ip g jq g kr g ls Rijkl ∇p u ∇q u ∇r u ∇s u = 0
a.e. in D ,
(4)
where Rijkl is the Riemannian curvature tensor of M .
Note that when M = Rn , Rijkl ≡ 0 ; hence the equality (4) holds automatically in this case. ¿From equality (4), we have ∇u = 0 at any point where
the curvature is positive (or negative). So we have:
EJDE–1995/07
Nobumitsu Nakauchi
3
Corollary 1 Suppose that the sectional curvature of M is positive (or negative)
2,2+ε
in D . Then any Wloc
-solution of (3) in D is a constant function.
We mention a related fact on harmonic functions. Let u be a harmonic
function on a Riemannian manifold M . Then u is a constant function if one
of the following two conditions holds:
1. M is compact (the maximum principle).
2. M is complete and non-compact, the Ricci curvature of M is nonnegative, and u is bounded on M (Yau [7]).
These results need the assumption that u is globally defined on compact or
complete manifolds. On the other hand, the above equality (4) holds when
an ∞-harmonic function u is defined on a subdomain of M ; the structure of
∞-Laplacian gives a restriction on local existence of solutions.
The author thinks that our theorem holds without the assumption that
2,2+ε
solutions belong to the class Wloc
(D) , though we use this assumption. Then
Aronsson’s minimization approach of the Lipschitz extention problem does not
seem to work on any positively (or negatively) curved manifold.
2
A Bochner type formula
In this section we prove the following formula of Bochner type.
Lemma 1 Let u be a C3loc -solution of (3) on a subdomain D of a Riemannian
manifold M . Then the following equality holds.
1
k∇k∇uk2 k2
2
+2 g ip g jq g kr g ls Rikjl ∇p u ∇q u ∇r u ∇s u
g ip g jq ∇i u ∇j u ∇p ∇q k∇uk2 +
(5)
= 0
in D ,
where k∇uk2 = g ij ∇i u∇j u and k∇k∇uk k2 = g ij ∇i k∇uk∇j k∇uk .
Proof. Note ∇gij = ∇g ij = 0 , since ∇ is the Levi-Civita connection.
Applying ∇r to both sides of (3), we have
g ip g jq ∇i u ∇j u ∇r ∇p ∇q u + 2 g ip g jq ∇i u ∇r ∇j u ∇p ∇q u = 0 .
(6)
We see that
∇p ∇q ∇r u
= ∇p ∇r ∇q u
= ∇r ∇p ∇q u − g ls Rprqs ∇l u (by the Ricci formula) .
(7)
∞-harmonic Functions
4
EJDE–1995/07
We get
g ip g jq g kr ∇i u ∇j u ∇p ∇k u ∇q ∇r u
1
1
= g kr ∇k (g ip ∇i u∇p u) ∇r (g jq ∇j u∇q u)
2
2
1 kr
2
g ∇k k∇uk ∇r k∇uk2
=
4
1
k∇k∇uk2 k2 .
=
4
(8)
Then we have
g ip g jq ∇i u ∇j u ∇p ∇q k∇uk2
= g ip g jq ∇i u ∇j u ∇p ∇q (g kr ∇r u∇k u)
= 2 g ip g jq g kr ∇i u ∇j u ∇p ∇q ∇r u ∇k u
+2 g ip g jq g kr ∇i u ∇j u ∇q ∇r u ∇p ∇k u
= 2 g ip g jq g kr ∇i u ∇j u ∇r ∇p ∇q u ∇k u
−2 g ip g jq g kr ∇i u ∇j u g ls Rprqs ∇l u ∇k u
+2 g ip g jq g kr ∇i u ∇j u ∇q ∇r u ∇p ∇k u
= − 4 g ip g jq g kr ∇i u ∇k u ∇r ∇j u ∇p ∇q u
−2 g ip g jq g kr g ls Rprqs ∇i u ∇j u ∇k u ∇l u
+2 g ip g jq g kr ∇i u ∇j u ∇q ∇r u ∇p ∇k u
= − 2 g ip g jq g kr g ls Rpqrs ∇i u ∇j u ∇k u ∇l u
−2 g ip g jq g kr ∇i u ∇j u ∇q ∇r u ∇p ∇k u
(by (7) )
(by (6) )
(by exchange of indices )
1
= − 2 g ip g jq g kr g ls Rpqrs ∇i u ∇j u ∇k u ∇l u − k∇k∇uk2 k2
(by (8) ) .
2
3
Proof of Theorem 1 for C3loc-solutions
Take any η ∈ C∞
0 (D) . Then from (5), we have
Z
Z
1
η g g ∇i u ∇j u ∇p ∇q k∇uk +
k∇k∇uk2 k2 η
2 D
Z
η g ip g jq g kr g ls Rikjl ∇p u ∇q u ∇r u ∇s u = 0 .
+2
ip jq
D
2
(9)
D
Note here
g jq ∇j u ∇q k∇uk2
= g jq ∇j u ∇q (g ip ∇i u∇p u)
= 2 g ip g jq ∇j u ∇i u ∇q ∇p u = 0 .
(10)
EJDE–1995/07
Nobumitsu Nakauchi
5
Using integration by parts, we get
Z
η g ip g jq ∇i u ∇j u ∇p ∇q k∇uk2
D
Z
= −
g ip g jq ∇p η ∇i u ∇j u ∇q k∇uk2
D
Z
−
η g ip g jq ∇p ∇i u ∇j u ∇q k∇uk2
D
Z
−
η g ip g jq ∇i u ∇p ∇j u ∇q k∇uk2
ZD
= −
η g ip g jq ∇i u ∇p ∇j u ∇q k∇uk2
D
Z
1
= −
η g jq ∇j (g ip ∇i u ∇p u) ∇q k∇uk2
2
D
Z
1
= −
η g jq ∇j k∇uk2 ∇q k∇uk2
2 D
Z
1
= −
k∇k∇uk2 k2 η .
2 D
(11)
(by (10) )
¿From (9) and (11), we have
Z
η g ip g jq g kr g ls Rikjl ∇p u ∇q u ∇r u ∇s u = 0 .
(12)
D
Since η is an arbitrary test function in C∞
0 (D) , we have
g ip g jq g kr g ls Rikjl ∇p u ∇q u ∇r u ∇s u = 0
4
a.e. in D .
2
Proof of Theorem 1
In this section we complete our proof of Theorem 1 using an approximation. For
2,2+ε
any Wloc
-solution u of (3), we take an approximating sequence { u(ν) }∞
ν =1
⊂ C3loc (D) such that for any compact set K in D ,
2,2+ε
1. ϕ(ν) : = u(ν) − u approaches zero in Wloc
(D) as ν tends to infinity,
and
2. the Lipschitz constants of u(ν) (ν = 1, 2, ...) are uniformly bounded
on K : hence k∇u(ν) kL∞ (K) and k∇ϕ(ν) kL∞ (K) (ν = 1, 2, ...) are
uniformly bounded on K .
Since u = u(ν) − ϕ(ν) satisfies (3), we have
g ip g jq ∇i (u(ν) − ϕ(ν) ) ∇j (u(ν) − ϕ(ν) ) ∇p ∇q (u(ν) − ϕ(ν) ) = 0
∞-harmonic Functions
6
EJDE–1995/07
i.e.,
g ip g jq ∇i u(ν) ∇j u(ν) ∇p ∇q u(ν) + F (ϕ(ν) , u(ν) ) = 0
(13)
where
F (ϕ(ν) , u(ν) )
= −g ip g jq ∇i ϕ(ν) ∇j u(ν) ∇p ∇q u(ν) − g ip g jq ∇i u(ν) ∇j ϕ(ν) ∇p ∇q u(ν)
−g ip g jq ∇i u(ν) ∇j u(ν) ∇p ∇q ϕ(ν) + g ip g jq ∇i ϕ(ν) ∇j ϕ(ν) ∇p ∇q u(ν)
+g ip g jq ∇i u(ν) ∇j ϕ(ν) ∇p ∇q ϕ(ν) + g ip g jq ∇i ϕ(ν) ∇j u(ν) ∇p ∇q ϕ(ν)
−g ip g jq ∇i ϕ(ν) ∇j ϕ(ν) ∇p ∇q ϕ(ν) .
Let ψ ∈ W01,1 (D) . Multiply by − ∇r ψ both sides of (13) and use integration
by parts, then we have
Z
ψ g ip g jq ∇i u(ν) ∇j u(ν) ∇r ∇p ∇q u(ν)
ZD
+2
ψ g ip g jq ∇i u(ν) ∇r ∇j u(ν) ∇p ∇q u(ν)
D
Z
−
F (ϕ(ν) , u(ν) ) ∇r ψ = 0 .
M
Let ψ = η g kr ∇k u(ν) and sum them up with respect to r . Then we get
Z
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇r ∇p ∇q u(ν) ∇k u(ν)
(14)
D
Z
+2
η g ip g jq g kr ∇i u(ν) ∇k u(ν) ∇r ∇j u(ν) ∇p ∇q u(ν)
D
Z
−
F (ϕ(ν) , u(ν) ) g kr ∇r (η ∇k u(ν) ) = 0
M
We see
Z
η g ip g jq ∇i u(ν) ∇j u(ν) ∇p ∇q k∇u(ν) k2
D
Z
=
η g ip g jq ∇i u(ν) ∇j u(ν) ∇p ∇q (g kr ∇r u∇k u)
D
Z
= 2
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇p ∇q ∇r u(ν) ∇k u(ν)
D
Z
+2
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇q ∇r u(ν) ∇p ∇k u(ν)
D
Z
= 2
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇r ∇p ∇q u(ν) ∇k u(ν)
D
Z
−2
η g ip g jq g kr ∇i u(ν) ∇j u(ν) g ls Rprqs ∇l u(ν) ∇k u(ν)
D
EJDE–1995/07
Nobumitsu Nakauchi
7
Z
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇q ∇r u(ν) ∇p ∇k u(ν)
+2
(by (7) )
Z
D
= −4
η g ip g jq g kr ∇i u(ν) ∇k u(ν) ∇r ∇j u(ν) ∇p ∇q u(ν)
ZD
F (ϕ(ν) , u(ν) ) g kr ∇k (η ∇r u(ν) )
+2
Z
D
−2
η g ip g jq g kr g ls Rprqs ∇i u(ν) ∇j u(ν) ∇k u(ν) ∇l u(ν)
Z
D
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇q ∇r u(ν) ∇p ∇k u(ν)
+2
(by (14) )
Z
D
= −2
η g ip g jq g kr g ls Rpqrs ∇i u(ν) ∇j u(ν) ∇k u(ν) ∇l u(ν)
Z
D
−2
η g ip g jq g kr ∇i u(ν) ∇j u(ν) ∇q ∇r u(ν) ∇p ∇k u(ν)
Z
D
F (ϕ(ν) , u(ν) ) g kr ∇k (η ∇r u(ν) )
+2
(by exchange of indices)
Z
D
= −2
η g ip g jq g kr g ls Rpqrs ∇i u(ν) ∇j u(ν) ∇k u(ν) ∇l u(ν)
Z
Z
1
(ν) 2 2
−
k∇k∇u k k + 2
F (ϕ(ν) , u(ν) ) g kr ∇k (η ∇r u(ν) )
2 D
D
D
(by (8) ) .
Therefore we obtain an integral form of the Bochner equality for u(ν) :
Z
Z
1
ip jq
(ν)
(ν)
(ν) 2
η g g ∇i u ∇j u ∇p ∇q k∇u k +
k∇k∇u(ν) k2 k2 η
2 M
M
Z
−2
F (ϕ(ν) , u(ν) ) g kr ∇k (η ∇r u(ν) )
(15)
M
Z
+2
η g ip g jq g kr g ls Rikjl ∇p u(ν) ∇q u(ν) ∇r u(ν) ∇s u(ν) = 0 .
M
for any η ∈ C∞
0 (D) . Note here
g jq ∇j u(ν) ∇q k∇u(ν) k2
= g jq ∇j u(ν) ∇q (g ip ∇i u(ν) ∇p u(ν) )
= 2 g g ∇j u
ip jq
= − 2 F (ϕ
(ν)
(ν)
,u
Then using integration by parts, we get
Z
η g ip g jq ∇i u(ν) ∇j u(ν) ∇p ∇q k∇u(ν) k2
D
Z
= −
g ip g jq ∇p η ∇i u(ν) ∇j u(ν) ∇q k∇u(ν) k2
D
∇i u
(ν)
)
(ν)
∇q ∇p u
(16)
(ν)
(by (13) ) .
(17)
∞-harmonic Functions
8
EJDE–1995/07
Z
−
η g ip g jq ∇p ∇i u(ν) ∇j u(ν) ∇q k∇u(ν) k2
Z
−
η g ip g jq ∇i u(ν) ∇p ∇j u(ν) ∇q k∇u(ν) k2
ZD
Z
= 2
g ip ∇p η ∇i u(ν) F (ϕ(ν) , u(ν) ) = 2
η g ip ∇p ∇i u(ν) F (ϕ(ν) , u(ν) )
D
D
Z
−
η g ip g jq ∇i u(ν) ∇p ∇j u(ν) ∇q k∇u(ν) k2
(by (16) )
D
Z
Z
g ip ∇i u(ν) ∇p η F (ϕ(ν) , u(ν) ) + 2
η 4 u(ν) F (ϕ(ν) , u(ν) )
= 2
D
D
Z
1
(ν)
2
−
k∇k∇u k k η ,
2 D
D
because
g ip ∇i u(ν) ∇p ∇j u(ν) =
1
1
∇j (g ip ∇i u(ν) ∇p u(ν) ) = ∇j k∇u(ν) k2 .
2
2
Then by (15) and (17), we obtain
Z
2
η g ip g jq g kr g ls Rijkl ∇p u(ν) ∇q u(ν) ∇r u(ν) ∇s u(ν)
D
Z
= −2
g ip ∇i u(ν) ∇p η F (ϕ(ν) , u(ν) )
D
Z
−2
η 4 u(ν) F (ϕ(ν) , u(ν) )
D
Z
+2
g kr ∇r (η ∇k u(ν) ) F (ϕ(ν) , u(ν) ) .
D
Since k∇u
(ν)
k is bounded uniformly on K , we get
| the right hand side of (18) |
Z
Z
≤ C
kF (ϕ(ν) , u(ν) )k + C
k∇∇u(ν) k kF (ϕ(ν) , u(ν) )k
K
K
Z
Z
(ν)
(ν)
≤ C
k∇ϕ k k∇∇u k + C
k∇∇ϕ(ν) k
K
K
Z
Z
+C
k∇ϕ(ν) k2 k∇∇u(ν) k + C
k∇ϕ(ν) k k∇∇ϕ(ν) k
K
K
Z
Z
(ν) 2
(ν)
+C
k∇ϕ k k∇∇ϕ k + C
k∇ϕ(ν) k k∇∇u(ν) k2
K
K
Z
Z
(ν)
(ν)
+C
k∇∇ϕ k k∇∇u k + C
k∇ϕ(ν) k2 k∇∇u(ν) k2
K
ZK
(ν)
(ν)
(ν)
+C
k∇ϕ k k∇∇ϕ k k∇∇u k
K
(18)
EJDE–1995/07
Nobumitsu Nakauchi
9
Z
k∇ϕ(ν) k2 k∇∇ϕ(ν) k k∇∇u(ν) k
+C
K
= : I1 + I2 + I3 + I4 + I5 + I6 + I7 + I8 + I9 + I10 .
Since ϕ(ν) converges to in W2,2+ε (K) as ν tends to infinity, ∇ϕ(ν) and
∇∇ϕ(ν) approaches zero in L2 (K) . Then
Z
1/2 Z
1/2
(ν) 2
(ν) 2
I1 ≤ C
k∇ϕ k
k∇∇u k
→ 0,
K
K
Z
1/2 Z
I2 ≤ C
k∇∇ϕ
1/2
k
(ν) 2
1
K
I3 ≤ C sup k∇ϕ
k
k∇ϕ
1/2
k
K
1/2 Z
k∇∇ϕ(ν) k2
K
→ 0,
K
Z
1/2 Z
I5 ≤ C sup k∇ϕ(ν) k
1/2
k∇ϕ(ν) k2
K
k∇∇ϕ(ν) k2
K
→ 0,
K
Z
1/2 Z
1/2
k∇∇ϕ(ν) k2
k∇∇u(ν) k2
K
→ 0,
K
Z
I9 ≤ C sup k∇ϕ
→ 0,
1/2
k∇ϕ(ν) k2
I7 ≤ C
k
K
Z
I4 ≤ C
(ν)
1/2 Z
k
k∇∇ϕ
k
k∇∇u
(ν) 2
K
K
k
(ν) 2
12
1/2 Z
k
→ 0,
K
Z
k∇∇ϕ
(ν) 2
K
k∇∇u
(ν) 2
1/2 Z
(ν) 2
K
I10 ≤ C sup k∇ϕ
→ 0,
K
Z
(ν)
2
1/2
k
k∇∇u
(ν) 2
K
k
(ν) 2
→ 0.
K
Furthermore, since ϕ(ν) converges to zero in W2,2+ε (K) , we have
Z
ε/(2+ε) Z
2/(2+ε)
(ν) 1−ε
(ν) 2+ε
(ν) 2+ε
I6 ≤ C sup k∇ϕ k
k∇ϕ k
k∇∇u k
→ 0,
K
K
K
Z
I8 ≤ C sup k∇ϕ
ε/(2+ε) Z
k
k∇ϕ
(ν) 2−ε
K
k
K
2/(2+ε)
k∇∇u
(ν) 2+ε
k
(ν) 2+ε
→ 0.
K
Thus the right hand side of (18) converges to zero as ν tends to infinity. Then,
letting ν go to infinity in (18), we have (12). This completes the proof.
2
10
∞-harmonic Functions
EJDE–1995/07
References
[1] Aronsson, G., Extension of function satisfying Lipschitz conditions, Ark.
Mat. 6 (1967), 551-561.
[2] Aronsson, G., On the partial differential equation u2x uxx + 2 ux uy uxy +
u2y uyy = 0 , Ark. Mat. 7 (1968), 395-425.
[3] Aronsson, G., On certain singular solutions of the partial differential equation u2x uxx + 2 ux uy uxy + u2y uyy = 0 , Manuscripta Math. 47 (1984),
131-151.
[4] Bhattacharya, T., DiBenedetto, E. & Manfredi, J., Limits as p → ∞ of
4p up = f and related extremal problems, Rend. Sem. Mat. Univ. Pol.
Torino, Fascicolo Speciale 1989 Nonlinear PDE’s, 15-68.
[5] Evans, L.C., Estimates for smooth absolutely minimizing Lipschitz extensions, Electronic J. Diff. Equations 1993 (1993), No.3, 1-9.
[6] Jensen, R., Uniqueness of Lipschitz extensions: minimizing the sup norm
of the gradient, Arch. Rational Mech. Anal. 123 (1993), 51-74.
[7] Yau, S.-T., Harmonic functions on complete Riemannian manifolds,
Comm. Pure Appl. Math. XXVIII (1975), 201-228.
Nobumitsu Nakauchi
Department of Mathematics
Faculty of Science
Yamaguchi University
Yamaguchi 753, Japan
E-mail: nakauchi@ccy.yamaguchi-u.ac.jp
EJDE–1995/07
Nobumitsu Nakauchi
11
December 11, 1996 Addendum
In this article, Theorem 1 follows from general properties of the Riemannian
curvature tensor, and Corollary 1 is incorrect. The Bochner formula does not
seem to work in this situation.
Lemma 1 can be used in proving the following Liouville theorem for C3 solutions.
Theorem A. Let M be a complete noncompact Riemannian manifold of nonnegative (sectional) curvature. Let u be a bounded ∞-harmonic function of
C3 -class on M . Then u is a constant function.
The curvature assumption in Theorem A is necessary only for applying the
Hessian comparison theorem in the proof (Here we use the operator Q ij =
g ip g jq ∇p ∇q ). Theorem A also follows from arguments in [Cheng, S.Y., Liouville
theorem for harmonic maps, Proc. Symp. Pure Math. 36(1980), 147-151].
See also the article [Hong, N.C., Liouville theorems for exponentially harmonic
functions on Riemannian manifolds, Manuscripta Math. 77(1992), 41-46].
Sincerely yours,
Nobumitsu Nakauchi