Energy finite solutions of elliptic equations on Riemannian manifolds

ENERGY FINITE SOLUTIONS OF ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

Seok Woo Kim and Yong Hah Lee

Reprinted from the

Journal of the Korean Mathematical Society Vol. 45, No. 3, May 2008

c

°2008 The Korean Mathematical Society

ENERGY FINITE SOLUTIONS OF ELLIPTIC EQUATIONS ON RIEMANNIAN MANIFOLDS

Seok Woo Kim and Yong Hah Lee

Abstract. We prove that for any continuous function f on the s-harmon- ic (1 < s < ∞) boundary of a complete Riemannian manifold M , there exists a solution, which is a limit of a sequence of bounded energy finite solutions in the sense of supremum norm, for a certain elliptic operator A on M whose boundary value at each s-harmonic boundary point coincides with that of f .

If E

, E

, . . . , E

are s-nonparabolic ends of M , then we also prove that there is a one to one correspondence between the set of bounded energy finite solutions for A on M and the Cartesian product of the sets of bounded energy finite solutions for A on E

which vanish at the boundary ∂E

for i = 1, 2, . . . , l.

1. Introduction

In 1975, Yau [13] proved that every positive harmonic function on a com- plete Riemannian manifold with nonnegative Ricci curvature must be constant.

Later, Li-Tam [9] studied various spaces of harmonic functions on a complete Riemannian manifold with nonnegative sectional curvature outside a compact set. In particular, Li-Tam proved that the dimension of the space of positive harmonic functions on such a manifold is equal to the number of ends of the manifold. Such a theory is well developed by works of Donnelly [1], Grigor’yan [2], Holopainen [5], Li-Tam [10], Sung-Tam-Wang [12], present authors [6], [7], and others.

The main purpose of this paper is to prove that bounded energy finite solu- tions of a nonlinear elliptic operator on a complete Riemannian manifold can be represented in terms of bounded energy finite solutions of such an operator on each end of the manifold. In particular, we give a one to one correspondence between the totality of bounded energy finite solutions of a nonlinear elliptic operator on a complete Riemannian manifold and the Cartesian product of

Received October 17, 2006.

2000 Mathematics Subject Classification. Primary 58J05, 31B05.

Key words and phrases. s-harmonic boundary, A-harmonic function, end.

The first author was supported by grant No. R01-2006-000-10047-0(2006) from the Basic Research Program of the Korea Science & Engineering Foundation.

c

°2008 The Korean Mathematical Society

807

those on ends of the manifold. In this paper, we consider a nonlinear elliptic operator A of type s in the following setting:

Let Ω be an open subset of a complete Riemannian manifold M and W ^1,s (Ω), (1 < s < ∞) be the Sobolev space of all functions u ∈ L ^s (Ω) whose distribu- tional gradient ∇u also belongs to L ^s (Ω). We equip W ^1,s (Ω) with the norm kuk 1,s = kuk s + k∇uk s . The subspace W ₀ ^1,s (Ω) is the closure of C ₀ ^∞ (Ω) in W ^1,s (Ω). We shall pay attention to functionals associated with F : T Ω → R, where

(A1) the mapping F x = F| T

M : T x M → R is strictly convex and differ- entiable for all x ∈ Ω, and the mapping x 7→ F x (ξ) is measurable whenever ξ is;

(A2) there exist constants 0 < C 1 ≤ C 2 < ∞ such that C 1 |ξ| ^s ≤ F x (ξ) ≤ C 2 |ξ| ^s for all x ∈ Ω and ξ ∈ T x M .

Let A x (ξ) = (A ¹ (ξ), A ² (ξ), . . . , A ⁿ (ξ)) be defined by A ⁱ (ξ) = ∂

∂ξ ⁱ F x (ξ) for i = 1, 2, . . . , n.

Then A also satisfies the following properties: (See [11].)

(A3) the mapping A x = A| T

M : T x M → T x M is continuous for a.e. x ∈ Ω, and the mapping x 7→ A x (ξ) is a measurable vector field whenever ξ is;

for a.e. x ∈ Ω and for all ξ ∈ T x M (A4) hA x (ξ), ξi ≥ C 1 |ξ| ^s ; (A5) |A x (ξ)| ≤ C 2 |ξ| ^s−1 ;

(A6) hA x (ξ 1 ) − A x (ξ 2 ), ξ 1 − ξ 2 i > 0 whenever ξ 1 6= ξ 2 .

A function u in W _loc ^1,s (Ω) is a solution (supersolution, subsolution, respectively) of the equation

(1) −divA x (∇u) = 0 (≥ 0, ≤ 0, respectively)

in Ω if Z

Ω

hA _x (∇u), ∇φi = 0 (≥ 0, ≤ 0, respectively)

for any (nonnegative, respectively) φ ∈ C ₀ ^∞ (Ω). A function u is said to be A-harmonic (of type s) if u is a continuous solution of the equation (1). In the typical case A x (ξ) = ξ|ξ| ^s−2 , A-harmonic functions are called s-harmonic and, in particular, if s = 2, we obtain harmonic functions. In fact, if we write

J(u, Ω) = Z

Ω

F x (∇u),

then given f ∈ W ^1,s (Ω), each A-harmonic function h with h − f ∈ W ₀ ^1,s (Ω)

minimizes the energy functional J in U = {u ∈ W ^1,s (Ω) : u − f ∈ W ₀ ^1,s (Ω)}.

(See [11].) In the case when an A-harmonic function u on Ω has finite energy, i.e., J(u, Ω) < ∞, we say that u is an energy finite A-harmonic function on Ω.

An important property is the comparison principle as follows: If u ∈ W ^1,s (Ω) is a supersolution and v ∈ W ^1,s (Ω) is a subsolution on an open set Ω and min{u − v, 0} ∈ W ₀ ^1,s (Ω), then u ≥ v a.e. in Ω. In particular, if both u and v are A-harmonic in a bounded set Ω and u ≥ v on ∂Ω, then u ≥ v in Ω. (See [3].) We now introduce additional assumptions on F as follows:

(A7) A x (λξ) = λ|λ| ^s−2 A x (ξ) whenever λ is a nonzero real number;

for any ξ ₁ , ξ ₂ ∈ T _x M (A8) in case 2 ≤ s < ∞,

F x ³ ξ 1 + ξ 2

2

´

+ F x ³ ξ 1 − ξ 2

2

´

≤ 1

2 (F x (ξ 1 ) + F x (ξ 2 ));

in case 1 < s ≤ 2, F x

³ ξ ₁ + ξ 2

2

´ _{s ˜} + F x

³ ξ ₁ − ξ 2

2

´ _{s ˜}

≤ ³ 1

2 (F x (ξ 1 ) + F x (ξ 2 ))

´ _{˜ s} , where ˜ s = 1/(s − 1).

Using Clarkson’s inequality, the assumption (A8) holds in the typical case F(ξ) = ¹ _s |ξ| ^s , i.e., the s-harmonic case. (See [4].)

In this paper, we solve the Dirichlet problem on the s-harmonic boundary of a complete Riemannian manifold as follows:

Theorem 1.1. Let M be a complete Riemannian manifold and A be an elliptic operator on M satisfying (A1), (A2), (A7), and (A8). Then for any continuous function f on the s-harmonic boundary ∆ M of M , there exists an A-harmonic function h on M , which is a limit of a sequence of bounded energy finite A- harmonic functions in the sense of supremum norm, such that for each x ∈ ∆ M ,

h(x) = f (x).

In particular, in the case that ∆ M consists of finitely many points x 1 , x 2 , . . ., x l , for given real numbers a 1 , a 2 , . . . , a l ∈ R, there exists a bounded energy finite A-harmonic function h on M such that

(2) h(x i ) = a i for i = 1, 2, . . . , l.

Conversely, each bounded energy finite A-harmonic function h on M is uniquely determined by the values in (2).

We also give a precise relation between the totality of all bounded energy finite A-harmonic functions on the manifold and the Cartesian product of those on its ends as follows:

Theorem 1.2. Let M be a complete Riemannian manifold and A be an el-

liptic operator on M satisfying (A1), (A2), (A7), and (A8). Let E 1 , E 2 , . . . , E l ,

l ≥ 1, be s-nonparabolic ends of M , whose s-harmonic boundaries have finite cardinality. Then there exists a one to one correspondence

Φ : HBD A (M ) → Y l i=1

HBD A (E i , ∂E i ),

where HBD A (X) and HBD A (X, ∂X) denote the set of bounded energy finite A-harmonic functions on X and the subset of elements of HBD A (X) vanishing at ∂X, respectively.

2. Preliminaries

Let M be a complete Riemannian manifold. Let BD s (M ) be the set of bounded continuous functions u whose distributional gradient ∇u belongs to L ^s (M ). Then BD s (M ) forms an algebra over the real number with the usual addition and multiplication of functions and scalar multiplication defined point- wise. We equip BD s (M ) with the following topology: For a sequence {f n } in BD s (M ) and a function f on M , we say that {f n } converges to f in BD s (M ) if {f n } is uniformly bounded on M , lim n→∞ sup _K |f n − f | = 0 for each compact subset K of M , and

n→∞ lim Z

M

|∇(f n − f )| ^s = 0.

We denote by BD s,0 (M ) the closure of the set of all compactly supported smooth functions in BD s (M ). It is easy to see that BD s,0 (M ) is not only a subalgebra but also an ideal of BD s (M ). We denote by HBD A (M ) the subset of all bounded energy finite A-harmonic functions in BD s (M ), where A is an elliptic operator on M satisfying (A1), (A2), (A7), and (A8).

For a complete Riemannian manifold M , there exists a locally compact Hausdorff space ˆ M , called the Royden s-compactification of M , which contains M as an open dense subset. In particular, every function f ∈ BD s (M ) can be extended to a continuous function, denoted again by f , on ˆ M and the class of such extended functions separates points in ˆ M . Moreover, ˆ M is unique up to a homeomorphism fixing M . We call the subset ∂ ˆ M = ˆ M \ M the Royden s-boundary of M . Throughout this paper, for a subset A of M , we denote by A the closure of A in ˆ ˆ M . We define the s-harmonic boundary of M by

∆ M = {x ∈ ∂ ˆ M : f (x) = 0 for all f ∈ BD s,0 (M )}.

M is said to be s-nonparabolic if M has positive s-capacity, i.e., there exists a compact subset K ⊂ M such that

Cap _s (K, ∞, M ) = inf

φ

Z

M

|∇φ| ^s > 0,

where the infimum is taken over all compactly supported smooth functions φ

with φ = 1 on K. Otherwise, M is called s-nonparabolic.

The following two lemmas give some useful properties related to bounded energy finite A-harmonic functions and s-harmonic boundary. (See [8] for the proof.)

Lemma 2.1. Let M be a complete Riemannian manifold. Then M is s- parabolic if and only if s-harmonic boundary ∆ M is empty.

Lemma 2.2. Let M be a complete s-nonparabolic Riemannian manifold. Then we have the followings:

(i) (Maximum principle) If h is a function in HBD A (M ) such that a ≤ h ≤ b on ∆ M

for some constants a ≤ b, then a ≤ h ≤ b on M .

(ii) (Duality relation) BD _s,0 (M ) = {f ∈ BD _s (M ) : f = 0 on ∆ _M }.

(iii) (Royden decomposition) For each f ∈ BD s (M ), there exist unique h ∈ HBD A (M ) and g ∈ BD s,0 (M ) such that f = h + g.

3. s-harmonic boundary and energy finite A-harmonic functions on ends

Let o be a fixed point of M . If n(r) denotes the number of unbounded components of M \ B r (o), then it is easy to prove that n(r) is nondecreasing in r > 0. Let lim r→∞ n(r) = k, where k may be infinity. Then we say that the number of ends of M is k. In particular, if k is finite, there exists r 0 > 0 such that n(r) = k for all r ≥ r ₀ . In this case, there exist mutually disjoint unbounded components E 1 , E 2 , . . . , E k of M \ B r

(o) and we call each E i an end of M . All ends are divided into two classes by the following s-parabolicity:

We say that E is s-nonparabolic if E has positive s-capacity, i.e., Cap _s (∂E, ∞, E) = inf

u

Z

E

|∇u| ^s > 0,

where the infimum is taken over all compactly supported smooth functions u with u = 1 on ∂E. Otherwise, E is called an s-parabolic end. If A is an elliptic operator on M satisfying (A1), (A2), (A7), and (A8), then by the quasi-minimizing property of A-harmonic functions, the s-nonparabolicity of E means that for some r 1 ≥ r 0 , there exists a continuous function u E , called an A-harmonic measure of E, on E such that

 



A u E = 0 in E \ B r

(o);

u E = 0 on B r

(o) ∩ E;

sup _E\B

r1

(o) u E = 1.

For an end E of M , we define the sets BD s (E) and BD s,0 (E) in the same

way as BD s (M ) and BD s,0 (M ), respectively. Let ˜ E be the Royden s-compact-

ification of E and ∂ ˜ E = ˜ E \ E be the Royden s-boundary of E. For a subset

A of E, we denote by ˜ A the closure of A in ˜ E. Then the s-harmonic boundary of E is the set

∆ E = {x ∈ ∂ ˜ E : f (x) = 0 for all f ∈ BD s,0 (E)}.

On the other hand, we define the set HBD A (E, ∂E), similarly as HBD A (M ), by the totality of bounded energy finite A-harmonic functions on E vanishing on ∂E.

Lemma 3.1. Let Ω be a subset of an end E of a complete Riemannian man- ifold. Suppose that ˜ Ω ∩ ∆ E = ∅. Then there exists a function w in BD s,0 (E) such that w > 1 on ˜ Ω.

Proof. If z ∈ ˜ Ω ∩ ∂ ˜ E, then z / ∈ ∆ E by the assumption. Thus by definition of

∆ E , there exists a function f ∈ BD s,0 (E) such that f (z) 6= 0. Let us define a function w z on ˜ E by

w z (x) = 2|f (x)|

|f (z)| .

Clearly w z is a nonnegative function in BD s,0 (E) satisfying w z (z) = 2. From the compactness of ˜ Ω ∩ ∂ ˜ E, there exist finitely many points z 1 , z 2 , . . . , z k in Ω ∩ ∂ ˜ ˜ E such that

Ω ∩ ∂ ˜ ˜ E ⊂ [ k j=1

{x ∈ ˜ E : w z

(x) > 1}.

Now suppose that z ∈ ˜ Ω \ ∪ ^k _j=1 {x ∈ ˜ E : w z

(x) > 1}. Define a function w z ∈ BD s (E) such that

 



Aw z = 0 in B 2 (z) \ B 1 (z);

w z = 2 on B 1 (z);

w z = 0 on E \ B 2 (z).

Then w z ∈ BD s,0 (E). From the compactness of ˜ Ω \ ∪ ^k _j=1 {x ∈ ˜ E : w z

(x) > 1}, there exist finitely many points z k+1 , z k+2 , . . . , z k+m in ˜ Ω \ ∪ ^k _j=1 {x ∈ ˜ E : w z

(x) > 1} such that

³ Ω \ ˜ [ k j=1

{x ∈ ˜ E : w z

(x) > 1}

´

⊂

k+m [

j=k+1

{x ∈ ˜ E : w z

(x) > 1}.

In particular, w = P _k+m

j=1 w z

belongs to BD s,0 (E) and w > 1 on ˜ Ω. ¤ We have the maximum principle on each end similarly as on the whole manifold as follows:

Lemma 3.2 (Maximum principle). Let E be an end of a complete Riemannian manifold and h be a bounded energy finite A-harmonic function defined on E which may not belong to HBD A (E, ∂E). Suppose that for some constants a ≤ b,

a ≤ h ≤ b on ∆ E ∪ ∂E.

Then a ≤ h ≤ b on E.

Proof. It suffices to prove that h ≤ b on E. Otherwise, we may assume that there exist a point x 0 ∈ E and a constant c such that h(x 0 ) > c > b. Let Ω be the component of {x ∈ E : h(x) > c} containing x 0 . Then ˜ Ω∩∆ E = ∅. Thus by Lemma 3.1, we can find a function w ∈ BD _s,0 (E) such that w > 1 on ˜ Ω. Hence there exists a sequence {g n } of compactly supported functions converging to w in BD s (E). We denote by w and g _n the symmetric extension of w| Ω and g n | Ω to Ω d , respectively, where Ω d denotes the double of Ω. Clearly, each g _n is compactly supported on Ω d , and {g _n } converges to w in BD s (Ω d ). Therefore, w ∈ BD s,0 (Ω d ) and w > 1 on Ω d . Since BD s,0 (Ω d ) is an ideal of BD s (Ω d ), 1 = (1/w)w ∈ BD s,0 (Ω d ). Hence there exists a sequence {φ n } of compactly supported functions converging to 1 in BD s (Ω d ). Then {hφ n } converges to h in BD s (Ω). Putting v = h − c, the function vφ n vanishes on ∂(Ω ∩ K n ), where K n is the support of φ n in Ω d . Hence we get

J(h, Ω) = Z

Ω

hA(∇h), ∇vi = lim

n→∞

Z

Ω

hA(∇h), ∇(vφ n )i = 0.

This implies that h ≡ c on Ω, which contradicts h(x 0 ) > c. ¤ We also have the duality relation between BD s,0 (E) and ∆ E as follows:

Lemma 3.3. Let E be an end of a complete Riemannian manifold. Then BD s,0 (E) = {f ∈ BD s (E) : f = 0 on ∆ E }.

Proof. Let f ∈ BD s (E) and f = 0 on ∆ E . Let {h n } n∈N be a sequence of continuous functions on M such that

 



A h n = 0 in B n (o) ∩ E;

h _n = 0 on ∂E;

h n = f on E \ B n (o).

Obviously, h n ∈ BD s (E) and |h n | ≤ sup _E |f | on E, hence the sequence {h n } is equicontinuous. By Ascoli’s theorem, there exists a subsequence {h n

} con- verging uniformly to a limit function h on any compact subset of E. Then h is an A-harmonic function by the result of [3].

On the other hand, there exists a constant α < ∞ such that α = inf

η J(f − η, E),

where the infimum is taken over all smooth functions η on E such that η is compactly supported on E with η = f on ∂E. By the minimizing property of A-harmonic functions, we get

n→∞ lim J(h n , E) = α.

In case 2 ≤ s < ∞, by (A8),

α ≤ J((h n + h m )/2, E)

≤ J((h n + h m )/2, E) + J((h n − h m )/2, E)

≤ 2 ⁻¹ (J(h n , E) + J(h m , E)) → α as n, m → ∞.

In case 1 < s ≤ 2, by (A8),

α ^{˜ s} ≤ J((h n + h m )/2, E) ^{˜ s}

≤ J((h n + h m )/2, E) ^{˜ s} + J((h n − h m )/2, E) ^{s ˜}

≤ (2 ⁻¹ (J(h n , E) + J(h m , E))) ^{s ˜} → α ^{s ˜} as n, m → ∞, where ˜ s = 1/(s − 1). These imply that

J(h n − h m , E) → 0 as n, m → ∞.

Hence, {h n

} converges to h in BD s (E). Since {f − h n

} is a sequence of compactly supported continuous functions and it converges to f −h in BD s (E), f − h ∈ BD s,0 (E). By Lemma 3.3 together with the definition of f , h = 0 on

∆ E . From Lemma 3.2, we get h ≡ 0 on E. This implies that f ∈ BD s,0 (E). ¤ Following the proof of Lemma 3.3, we get a decomposition theorem which is the end version of the Royden decomposition theorem as follows:

Proposition 3.4. For each f ∈ BD s (E), there exist unique h ∈ HBD A (E, ∂E) and g ∈ BD s,0 (E) such that f = h + g.

Proof. To prove the uniqueness of the decomposition, suppose that there exist h ⁰ ∈ HBD A (E, ∂E) and g ⁰ ∈ BD s,0 (E) such that f = h ⁰ + g ⁰ . Then since g ⁰ − g ∈ BD s,0 (E), h − h ⁰ ∈ BD s,0 (E). Hence there exists a sequence {φ n } of compactly supported continuous functions converging to h − h ⁰ in BD s (E) with φ n = 0 on ∂E. On the other hand, since h, h ⁰ ∈ HBD A (E, ∂E),

Z

E

hA(∇h), ∇(h − h ⁰ )i = 0 and Z

E

hA(∇h ⁰ ), ∇(h − h ⁰ )i = 0,

hence Z

E

hA(∇h) − A(∇h ⁰ ), ∇h − ∇h ⁰ )i

= Z

E

hA(∇h), ∇(h − h ⁰ )i − Z

E

hA(∇h ⁰ ), ∇(h − h ⁰ )i = 0.

By (A6), this implies that h − h ⁰ ≡ C on E for some constant C. Since h − h ⁰ is continuous and h − h ⁰ ≡ 0 on ∂E, C ≡ 0. Thus we have h ≡ h ⁰ on E, hence

also g ≡ g ⁰ on E. ¤

We give a characterization of the s-parabolicity in terms of s-harmonic boundary as follows:

Theorem 3.5. Let E be an end of a complete Riemannian manifold. Then E

is s-parabolic if and only if s-harmonic boundary ∆ E is empty.

Proof. Suppose that there exists a point x ∈ ∆ E . Let {u r } r≥r

be a sequence of continuous functions on E such that

 



A u r = 0 in E ∩ B r (o);

u r = 0 on ∂E;

u r = 1 on E \ B r (o).

Obviously, each u r is a function in BD s (E) with u r (x) = 1. Since the sequence {u r } is decreasing, it converges to a limit function u in BD s (E) such that

 



A u = 0 in E;

u = 0 on ∂E;

0 ≤ u ≤ 1 on E.

Since {u r

−u r } r>r

is a sequence of compactly supported functions in BD s (E), there exists a function g ∈ BD s,0 (E) such that u r

= u + g, hence by Lemma 3.3, u(x) = 1. This implies that u is an A-harmonic measure of E, which means that E is s-nonparabolic.

Conversely, let E be an s-nonparabolic end and u E be an A-harmonic mea- sure of E. Suppose that ∆ _E is empty. By Lemma 3.1, there exists a function w ∈ BD s,0 (E) such that w > 1 on ˜ E. Since BD s,0 (E) is an ideal of BD s (E), 1 = (1/w)w ∈ BD s,0 (E). Thus there exists a sequence {φ n } of compactly sup- ported functions converging to 1 in BD s (E). Clearly, {u E φ n } converges to u E

in BD s (E). Therefore, we have J(u E , E) =

Z

M

hA(∇u E ), ∇(u E )i = lim

n→∞

Z

M

hA(∇u E ), ∇(u E φ n )i = 0.

This implies that u E is constant on E, which is a contradiction. ¤ Now, we solve the Dirichlet problem on the s-harmonic boundary of each end, and we describe the set of all bounded energy finite A-harmonic functions on each end in terms of s-harmonic boundary as follows:

Theorem 3.6. Let E be an end of a complete Riemannian manifold and A be an elliptic operator on E satisfying (A1), (A2), (A7), and (A8). Then for any continuous function f on ∆ E , there exists an A-harmonic function h on E, which is a limit of a sequence of functions in HBD A (E, ∂E) in the sense of supremum norm, such that for any x ∈ ∆ _E ,

h(x) = f (x).

In particular, in the case that ∆ E consists of finitely many points x 1 , x 2 , . . ., x l , for given real numbers a 1 , a 2 , . . . , a l ∈ R, there exists an A-harmonic function h ∈ HBD A (E, ∂E) such that

(3) h(x i ) = a i for i = 1, 2, . . . , l.

Conversely, each A-harmonic function h ∈ HBD A (E, ∂E) is uniquely deter-

mined by the values in (3).

Proof. For any continuous function f on ∆ E , f can be extended to a continuous function, denoted again by f , on ˜ E. Since BD s (E) is dense in the set of continuous functions on ˜ E with respect to the supremum norm, there exists a sequence {f n } converging to f in BD s (E). By Proposition 3.4, there exist h n ∈ HBD A (E, ∂E) and g n ∈ BD s,0 (E) such that f n = h n + g n . Since g n = 0 on ˜ ∆ E , we have h n (x) = f n (x) for each x ∈ ∆ E . Since the sequence {h n } is uniformly bounded, it is equicontinuous, and by Ascoli’s theorem there exists a subsequence {h n

} converging uniformly to a limit function h on any compact subset of E. Then h is an A-harmonic function by the result of [3]. Since for any x ∈ ∆ E and ² > 0,

|h(x) − f (x)| ≤ |h(x) − h n

(x)| + |f n

(x) − f (x)| < 2², we have

h(x) = f (x), x ∈ ∆ E .

Since BD s (E) is dense in the set of continuous functions on ˜ E with respect to the supremum norm, we can find f 1 , f 2 , . . . , f l ∈ BD s (E) such that f i (x j ) = δ ij

for i, j = 1, 2, . . . , l, where δ ij denotes Kronecker’s delta. Let f = P _l

i=1 a i f i . Then f (x i ) = a i for each i = 1, 2, . . . , l. By Proposition 3.4, there exist h ∈ HBD A (E, ∂E) and g ∈ BD s,0 (E) such that f = h + g. Since g = 0 on ˜ ∆ E by Lemma 3.3, we have h(x i ) = a i for each i = 1, 2, . . . , l.

Suppose that ˜h is another A-harmonic function in HBD A (E, ∂E) satisfying (3). By Lemma 3.3, h − ˜h ∈ BD s,0 (E). Since there exists a sequence of compactly supported continuous functions converging to h − ˜h in BD s (E),

Z

E

hA(∇h), ∇(h − ˜h)i = 0 and Z

E

hA(∇˜h), ∇(h − ˜h)i = 0.

Arguing similarly as Proposition 3.4, we have h−˜h ≡ C on E for some constant C. However, since E is s-nonparabolic, C ≡ 0. Thus we have h ≡ ˜h on E. ¤ Slightly modifying the proof of Theorem 3.6, one can prove Theorem 1.1.

4. Proof of the main theorem

In this section, we give a precise relation between the totality of bounded energy finite A-harmonic functions on the whole manifold and those on its ends. We first prove that the set of all s-harmonic boundary points of the whole manifold coincides the union of the set of all s-harmonic boundary points of each end.

Theorem 4.1. Let M be a complete Riemannian manifold with s-nonparabolic ends E 1 , E 2 , . . . , E l , l ≥ 1. Then ∆ M ∩ ˆ E i = ∆ E

for each i = 1, 2, . . . , l,

∆ M = [ l i=1

∆ E

and ]∆ M = P _l

i=1 ]∆ E

, where ]X denotes the cardinality of the set X.

Proof. Let x be a point in ∆ M ∩ ˆ E i and f be a function in BD s,0 (E i ). There exists a sequence {f n } of compactly supported functions converging to f in BD s (E i ). Choose a smooth function η defined on E i in such a way that η = 1 in E i \ B r

(o) for some r 2 > r 1 , η = 0 on ∂E i , and |∇η| is bounded. For each n, let us extend ηf and ηf n outside E i with zero, respectively. Then {ηf n }is a sequence of compactly supported functions converging to ηf in BD s (M ) hence ηf is a function in BD s,0 (M ). By definition of ηf ,

x→∞,x∈E lim

(f − ηf )(x) = 0.

Since ηf ∈ BD s,0 (M ) and x ∈ ∆ M , ηf (x) = 0, hence f (x) = 0. Therefore, by Lemma 3.3, x ∈ ∆ E

.

Conversely, let x be a point in ∆ _E

and f be a function in BD _s,0 (M ).

There exists a sequence {f n } of compactly supported functions converging to f in BD s (M ). Then {f n | E

} is a sequence of compactly supported functions converging to f | E

in BD s (E i ), hence f | E

∈ BD s,0 (E). Since f | E

(x) = 0, we conclude that f (x) = 0. Hence by Lemma 2.2, x ∈ ∆ M ∩ ˆ E i .

If E is an s-parabolic end, then by Theorem 3.5, we have ∆ E = ∅. Combining this together with the above arguments, we have the rest equalities. ¤

We are ready to prove our main theorem:

Proof of Theorem 1.2. Let h be a function in HBD A (M ). Combining Theorem 1.1, Theorem 4.1, and Theorem 3.6, we can construct a unique function h i in HBD A (E i , ∂E i ) in such a way that

h(x) = h i (x), x ∈ ∆ E

for each i = 1, 2, . . . , l. Let us define Φ : HBD A (M ) → Q _l

i=1 HBD A (E i , ∂E i ) by

Φ(h) = (h 1 , h 2 , . . . , h l ).

Then by the uniqueness of the functions h 1 , h 2 , . . . , h l , the map Φ is well de- fined.

Suppose that Φ(h) = Φ(h ⁰ ) for some functions h and h ⁰ in HBD A (M ). Then by definition of Φ, we have h − h ⁰ ∈ BD s,0 (M ). Since there exists a sequence of compactly supported continuous functions converging to h − h ⁰ in BD s (M ),

Z

M

hA(∇h), ∇(h − h ⁰ )i = 0 and Z

M

hA(∇h ⁰ ), ∇(h − h ⁰ )i = 0.

Arguing similarly as Proposition 3.4, we have h − h ⁰ ≡ C on M for some constant C. However, since M is s-nonparabolic, C ≡ 0. Thus we have h ≡ h ⁰ on M . Hence the map Φ is injective.

Let (h 1 , h 2 , . . . , h l ) ∈ Q _l

i=1 HBD A (E i , ∂E i ). Extend h i outside E i with zero for each i = 1, 2, . . . , l. Then the function f = P _l

i=1 h i belongs to BD s (M ).

By Proposition 3.4, there exist h ∈ HBD A (M ) and g ∈ BD s,0 (M ) such that

f = h+g. Since g = 0 on ∆ M and f (x) = h i (x), x ∈ ∆ E

for each i = 1, 2, . . . , l, we have

h(x) = h i (x), x ∈ ∆ E

for each i = 1, 2, . . . , l. Hence Φ(h) = (h 1 , h 2 , . . . , h l ), i.e., the map Φ is surjec-

tive. ¤

Applying our argument to the case of harmonic functions, we have the follow- ing linear isomorphism between the space of bounded harmonic functions with finite Dirichlet integral on a complete Riemannian manifold and the Cartesian product of those on its ends:

Corollary 4.2. Let M be a complete Riemannian manifold with nonparabolic ends E 1 , E 2 , . . . , E l , l ≥ 1. Then there exists an isomorphism

Φ : HBD(M ) → Y l i=1

HBD(E i , ∂E i ),

where HBD(X) and HBD(X, ∂X) denote the space of bounded harmonic func- tions with finite Dirichlet integral on X and the subspace of elements of HBD(X) vanishing at ∂X, respectively.

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Seok Woo Kim

Department of Mathematics Education Konkuk University

Seoul 143-701, Korea

E-mail address: [email protected] Yong Hah Lee

Department of Mathematics Education Ewha Womans University

Seoul 120-750, Korea

E-mail address: [email protected]