on the boundary

POLONICI MATHEMATICI LXXIV (2000)

Harmonic functions in a cylinder with normal derivatives vanishing

on the boundary

by Ikuko Miyamoto and Hidenobu Yoshida (Chiba)

Dedicated to the memory of Professor Bogdan Ziemian

Abstract. A harmonic function in a cylinder with the normal derivative vanishing on the boundary is expanded into an infinite sum of certain fundamental harmonic functions.

The growth condition under which it is reduced to a finite sum of them is given.

1. Introduction. Let R

(n ≥ 2) denote the n-dimensional Euclidean space. The solution of the Neumann problem for an infinite cylinder

Γ

(D) = {(X, y) ∈ R

: X ∈ D, −∞ < y < ∞},

with D a bounded domain of R

, is not unique, because we can add to each solution harmonic functions in Γ

(D) with normal derivatives vanishing on the boundary. Hence, to classify general solutions we need to characterize such functions. If D = (0, π) and Γ

(D) is the strip

H = {(x, y) ∈ R

: 0 < x < π, −∞ < y < ∞},

then by applying a result of Widder [6, Theorem 2] which characterizes a harmonic function in H vanishing continuously on the boundary ∂H of H, we can obtain the following result:

Theorem A. Let h(x, y) be a harmonic function in H such that ∂h/∂x vanishes continuously on ∂H. Then

h(x, y) = A

y + B

+

∞

X

k=1

(A

e

+ B

e

) cos kx,

where the series converges for all x and y, and A

, B

, A

, B

, A

, B

, . . . are

2000 Mathematics Subject Classification: Primary 31B20.

Key words and phrases : Neumann problem, harmonic functions, cylinder.

[229]

constants such that

A

e

+ B

e

= 2 π

π

\

0

h(x, y)kx dx (k = 1, 2, . . .).

Although this theorem is easily proved, we cannot proceed similarly in the case where Γ

(D) is a cylinder in R

(n ≥ 3). This kind of problem was originally treated by Bouligand [1] in 1914.

Theorem B (Bouligand [1, p. 195]). Let h(X, y) be a harmonic function in Γ

(D) such that the normal derivative of h vanishes continuously on the boundary ∂Γ

(D) of Γ

(D). If h(X, y) tends to zero as |y| → ∞, then h(X, y) is identically zero in Γ

(D).

In this paper we shall prove a cylindrical version of Theorem A (The- orem). As corollaries we shall obtain two results generalizing Theorem B (Corollaries 1 and 2).

2. Preliminaries. Let D be a bounded domain in R

(n ≥ 3) having a sufficiently smooth boundary ∂D. For example, D can be a C

-domain (0 < α < 1) in R

bounded by a finite number of mutually disjoint closed hypersurfaces (see Gilbarg and Trudinger [3, pp. 88–89] for the definition of C

-domain). Consider the Neumann problem

(2.1) (∆

+ µ)ϕ(X) = 0

for any X = (x

, . . . , x

) ∈ D,

(2.2) lim

X→X^′, X∈D

(∇

ϕ(X), ν(X

)) = 0 for any X

∈ ∂D, where

∆

= ∂

∂x

+ . . . + ∂

∂x

, ∇

=

 ∂

∂x

, . . . , ∂

∂x

 and ν(X

) is the outer unit normal vector at X

∈ ∂D.

Let {µ

(D)}

be the non-decreasing sequence of non-negative eigen- values of this Neumann problem. In this sequence we write µ

(D) the num- ber of times equal to the dimension of the corresponding eigenspace. If the normalized eigenfunction corresponding to µ

(D) is denoted by ϕ

(D)(X), the set of consecutive eigenfunctions corresponding to the same value of µ

(D) in the sequence {ϕ

(D)(X)}

forms an orthonormal basis for the eigenspace of the eigenvalue µ

(D). It is evident that µ

(D) = 0 and

ϕ

(D)(X) = |D|

(X ∈ D), |D| =

\

D

dX.

In the following we shall denote {µ

(D)}

and {ϕ

(D)(X)}

by

{µ(k)}

and {ϕ

(X)}

respectively, without specifying D. For each D

there is a sequence {k

} of non-negative integers such that k

= 0, k

= 1, µ(k

) < µ(k

),

µ(k

) = µ(k

+ 1) = µ(k

+ 2) = . . . = µ(k

− 1)

and {ϕ

, ϕ

, . . . , ϕ

} is an orthonormal basis for the eigenspace of the eigenvalue µ(k

) (i = 0, 1, 2, . . .). Since D has a sufficiently smooth boundary, we know that

µ(k) ∼ A(D, n)k

(k → ∞) and

X

µ(k)≤t

{ϕ

(X)}

∼ B(D, n)t

(t → ∞)

uniformly with respect to X ∈ D, where A(D, n) and B(D, n) are constants depending on D and n (e.g. see Carleman [2], Minakshisundaram and Plei- jel [4], Weyl [5]). Hence there exist positive constants M

, M

such that

M

k

≤ µ(k) (k = 1, 2, . . .) and

|ϕ

(X)| ≤ M

k

(X ∈ D, k = 1, 2, . . .).

3. Statement of our results. The gradient of a function f (P ) defined on Γ

(D) is

∇

f (P ) =  ∂f

∂x

(P ), . . . , ∂f

∂x

(P ), ∂f

∂y (P )

 (P = (x

, . . . , x

, y) ∈ Γ

(D)). We first remark that

I

(P ) = e √

ϕ

(X) and J

(P ) = e

√

ϕ

(X) (P = (X, y) ∈ Γ

(D)) are harmonic functions on Γ

(D) satisfying

P →Q, P ∈Γ

lim

(∇

I

(P ), ν(Q)) = 0 and

P →Q, P ∈Γ

lim

(∇

J

(P ), ν(Q)) = 0, where ν(Q) is the outer unit normal vector at Q ∈ ∂Γ

(D).

Theorem. Let h(P ) be a harmonic function on Γ

(D) satisfying

(3.1) lim

P →Q, P ∈Γn(D)

(∇

h(P ), ν(Q)) = 0 for any Q ∈ ∂Γ

(D). Then

h(P ) = A

y + B

+

∞

X

k=1

(A

I

(P ) + B

J

(P ))

for any P = (X, y) ∈ Γ

(D), where the series converges uniformly and absolutely on any compact subset of the closure Γ

(D) of Γ

(D), and A

, B

(k = 0, 1, 2, . . .) are constants such that (3.2) A

e √

µ(k) y

+ B

e

√

µ(k) y

=

\

D

h(X, y)ϕ

(X) dX (k = 1, 2, . . .).

Corollary 1. Let p and q be non-negative integers. If h(P ) is a har- monic function on Γ

(D) satisfying (3.1) and

(3.3) lim

y→∞

e

√

µ(kp+1) y

M

(y) = 0, lim

y→−∞

e √

µ(kq+1) y

M

(y) = 0, where

M

(y) = sup

X∈D

|h(X, y)| (−∞ < y < ∞), then

h(P ) = A

y + B

+

kp+1−1

X

k=1

A

I

(P ) +

kq+1−1

X

k=1

B

J

(P )

for any P = (X, y) ∈ Γ

(D), where A

(k = 0, 1, . . . , k

− 1) and B

(k = 0, 1, . . . , k

− 1) are constants.

Corollary 2. Let h(P ) be a harmonic function on Γ

(D) satisfying (3.1) and

M

(y) = o(e √

) (|y| → ∞).

Then h(P ) = A

y + B

for any P = (X, y) ∈ Γ

(D), where A

and B

are constants.

4. Proofs of Theorem and Corollaries 1, 2. Let f (X, y) be a function on Γ

(D). The function c

(f, y) of y (−∞ < y < ∞) defined by

c

(f, y) =

\

D

f (X, y)ϕ

(X) dX

is simply denoted by c

(y) in the following, without specifying f .

Lemma 1. Let h(P ) be a harmonic function on Γ

(D) satisfying (3.1).

Then

c

(y) = A

y + B

, (4.1)

c

(y) = A

e √

µ(k) y

+ B

e

√

µ(k) y

(k = 1, 2, . . .) (4.2)

with constants A

, B

(k ≥ 0) and

c

(y) = {e √

µ(k) (y−y2)

− e √

µ(k) (y2−y)

}c

(y

) e √

µ(k) (y1−y2)

− e √

µ(k) (y2−y1)

(4.3)

+ {e √

µ(k) (y1−y)

− e √

µ(k) (y−y1)

}c

(y

) e √

1−y2)

− e √

2−y1)

for any y

and y

, −∞ < y

< y

< ∞ (k = 1, 2, 3, . . .).

P r o o f. First of all, we remark that h ∈ C

(Γ

(D)) (Gilbarg and Trudin- ger [3, p. 124]). Since

\

D

(∆

h(X, y))ϕ

(X) dX =

\

D

h(X, y)(∆

ϕ

(X)) dX (−∞ < y < ∞), from Green’s identity, (2.2) and (3.1), we have

∂

c

(y)

∂y

=

\

D

∂

h(X, y)

∂y

ϕ

(X) dX = −

\

D

∆

h(X, y)ϕ

(X) dX

= −

\

D

h(X, y)(∆

ϕ

(X)) dX

= µ(k)

\

D

h(X, y)ϕ

(X) dX = µ(k)c

(y)

from (2.1) (k = 0, 1, 2, . . .). With constants A

and B

(k = 0, 1, 2,. . .) these give

c

(y) = A

y + B

and

c

(y) = A

e √

µ(k) y

+ B

e

√

µ(k) y

(k = 1, 2, . . .),

which are (4.1) and (4.2). When we solve for A

and B

the equations c

(y

) = A

e √

µ(k) yi

+ B

e

√

µ(k) yi

(i = 1, 2), we immediately obtain (4.3).

Remark . From (4.2) we have, for k = 1, 2, . . .

y→∞

lim c

(y)e

√

= A

and lim

y→−∞

c

(y)e √

= B

.

Lemma 2. Let h(P ) be a harmonic function on Γ

(D) satisfying (3.1).

Let y be any number and y

, y

be two any numbers satisfying −∞ < y

<

y − 1, y + 1 < y

< ∞. For two non-negative integers p and q,

∞

X

k=kp+q+1

|c

(y)| · |ϕ

(X)| ≤ L(p)M

(y

) + L(q)M

(y

),

where

L(j) = M

|D|

∞

X

k=kj+1

k exp(− pM

k

).

P r o o f. From Lemma 1, we see that

c

(y) = exp{−pµ(k)(y − y

)} 1 − exp{2pµ(k)(y − y

)}

1 − exp{2 pµ(k)(y

− y

)} c

(y

) + exp{pµ(k)(y − y

)} 1 − exp{2 pµ(k)(y

− y)}

1 − exp{2 pµ(k)(y

− y

)} c

(y

).

Hence (4.4)

∞

X

k=kp+q+1

|c

(y)| · |ϕ

(X)| ≤ I

+ I

, where

I

=

∞

X

k=kp+1

exp{−pµ(k)(y − y

)}|c

(y

)| · |ϕ

(X)|

I

=

∞

X

k=kq+1

exp{− pµ(k)(y

− y)}|c

(y

)||ϕ

(X)|.

For I

, we have

I

≤ M

|D|M

(y

)

∞

X

k=kp+1

k exp(− pµ(k)) (4.5)

≤ M

|D|M

(y

)

∞

X

k=kp+1

k exp(− pM

k

), because y − y

> 1.

For I

, we also have

(4.6) I

≤ M

|D|M

(y

)

∞

X

k=kq+1

k exp(− pM

k

).

Finally (4.4)–(4.6) give the conclusion of the lemma.

Proof of Theorem. Take any compact set T ⊂ Γ

(D) and two numbers y

, y

satisfying

max{y : (X, y) ∈ T } + 1 < y

, min{y : (X, y) ∈ T } − 1 > y

.

Let (X, y) be any point in T . Since c

(y) is the Fourier coefficient of the

function h(X, y) of X with respect to the orthonormal sequence {ϕ

(X)}

,

we have

h(X, y) =

∞

X

k=0

c

(y)ϕ

(X)

where the series converges uniformly and absolutely on T by Lemma 2.

Further (4.1) and (4.2) of Lemma 1 give (3.2). The proof of the Theorem is complete.

Proof of Corollaries 1 and 2. From (3.3) and the Remark, it follows that A

= 0 for any k ≥ k

and B

= 0 for any k ≥ k

. Hence the Theorem immediately gives the conclusion of Corollary 1. By putting p = q = 0 in Corollary 1, we obtain Corollary 2 at once.

References

[1] M. G. B o u l i g a n d, Sur les fonctions de Green et de Neumann du cylindre, Bull.

Soc. Math. France 42 (1914), 168–242.

[2] T. C a r l e m a n, Propri´et´es asymptotiques des fonctions fondamentales des mem- branes vibrantes , C. R. Skand. Math. Kongress 1934, 34–44.

[3] D. G i l b a r g and N. S. T r u d i n g e r, Elliptic Partial Differential Equations of Second Order, Springer, 1977.

[4] S. M i n a k s h i s u n d a r a m and ˚ A. P l e i j e l, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, Canad. J. Math. 1 (1949), 242–256.

[5] H. W e y l, Das asymptotische Verteilungsgestez der Eigenwerte linearer partieller Dif- ferentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann. 71 (1912), 441–479.

[6] D. V. W i d d e r, Functions harmonic in a strip, Proc. Amer. Math. Soc. 12 (1961), 67–72.

Department of Mathematics and Informatics Chiba University

Chiba 263-8522, Japan

E-mail: miyamoto@math.s.chiba-u.ac.jp yoshida@math.s.chiba-u.ac.jp

Re¸ cu par la R´ edaction le 25.4.1999