The Dirichlet problem for Laplace’s equation and for unbounded and half-unbounded 2-dimensional strip

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria l: PRACE MATEMATYCZNE XXV (1985)

M

F

and J

M

(Krakôw)

The Dirichlet problem for Laplace’s equation

and for unbounded and half-unbounded 2-dimensional strip

1. In the present paper we shall construct the solution и of the Laplace equation

(1) Au(X) = 0, X = (xu x 2)

in the domain

D = {X: |xj| < oo, |x2| < a}, where a is a positive constant, with boundary conditions

(2) u ( x u - a ) = / 1(x1), u{xu a) = f

(x1) for х г е Я = ( - с о , oo) and the solution v of equation (1) in the set

£>! = {X: Xj > 0, |x2| < a]

with the boundary conditions

(3) v (x 1, - a ) = g

( x1), v ( x

, a ) = g

{xd f o r x j e ^ c o ) and

(4) y(0, x2) = 03(x2) for x

e ( - a , a ) .

The functions f t (i = 1, 2), g( {i — 1, 2, 3) are given functions. To the solving of these two problems we shall use the convenient auxiliary functions Gf (i

= 1, 2, 3) which are the derivatives of the summands of the Green functions obtained by symmetric images. The solution of Dirichlet problem for the strip was obtained in papers [1], [3], [5], [6] by the Fourier convolution and conformal mapping.

Moreover, in paper [ 2] it has been solved the boundary problem for equation (1) in the domain D with the boundary conditions

u{xu - a ) = h{x1), DX

u{xl ,a) = 0 f o r x j e K

by using the Fourier’s convolution and conformal mapping (similar as in [1], V I [5], [6]).

Roczniki PTM — Prace Matematyczne XXV

2. Let X e D and Y = {yu y 2) denote a point of the set D**{Y: Ы < o o , W O } , Г Ф Х . Let

r„,- = U ,(X , Y ) = ф у г - х ^ М Ъ , , ) 2 {i = 1 ,2 ; n = 0, 1, ...), where

dnA = y

+ ( - l ) n+

[_x

+

na], d

= y 2+ ( - 1)"+1 [ x i - l n a ] for n — 0, 1, 2, ...

Let us now take into consideration the functions

00

Gl { X , y 1) = {Dy [ i n r

t l - ] n r ltlli+ £ ^ У

[ \ п г

пЛ- \ п г 2п+1Л + n — 1

+ 1° Г

п,

~^П r 2n-l,2]}|y2=

00

G2(2f, )^i) = }Dy2[ln r 0)2- l n r 1>2] + £ £>У2[1п r2n,2- l n r

+ n= 1

+ ln r 2(Itl- l n г2я_ 1,1]}|У2 = в,

00

G

{X, y 2) = \Dn [In r0>1- l n r 1>2 + X ( - 1 ) " ^ ! Dn 4 i - M=1

- I n r„+1>2]}|y1 = 0 = [Dy i [ln r0tl- l n r ltl] +

00

+ Z Cln rn>2 —In Гя+1.1] } |, 1 = 0 - n “ 1

Since /)У2 In rnJ = d„j(r„ti)~2, Dyi In r„if = {yx - x ^ r ^ y 2, by using the connections

^2и,1 = — d2n+ 1д = — а — х2 — dan (п = 0, 1,

^2п,2 = — d

n- i

~ —а — х

+ Аап (и = 1, 2, ...) for у2 = - а d-

n

— d

n+1,2 = а — х

+ 4ап (п = 0, 1,

d-

n, 1 = d2n— i,i = а - * 2- 4 а и (п = 1, 2, ...) for у2 = а, the functions G, (i = 1, 2, 3) may be written in the form

( 5 .)

Gt (X, у,) = 2Л„д + 2 £ И „ л + ^ . 2)

л = 1

00

= 2Лол + 2 А

+ 2 Y, И«,1+^11+1,2)»

и= 1

(

5

G2(X ,y ,) = 2B„,2 + 2 £ (B„,2 + B„,,) Л = 1

00

= 2S0t2 + 2B1>1 + 2 ]T (^ n,2 + £„+ i,i)>

n= 1

(53)

where

G A X , y 2) = CQA- C U2+ £ ( - 1 Г ( С „ д - С „ +1>2) n= 1

= С0Л- С М + I (-1)"(С „>2- С П+1Л), n= 1

( 6 )

^

.

= (2f, >4) =

A

= ^«,2(2^, У1) =

B„,2 — Bn2 (2^? У1) =

B„1 = £„д (2f, j/J =

■a — x-, —4 an

Q i

, y 2) — ^ ч2

Q 2 = Cna(X, y 2)

(.Vi ~ * i)2 + ( ~ « - x2 - 4an

— a — x

+ 4an (>>1 —x

)

jr{ — a — x

+ 4an

a — x

+ 4an {y ! —x 1)2+(a — x

+ 4ari

a — x

— 4an (У1 - x

+ { a - x

— 4 an

- * 1 (*i)2 + « i ) 2

- * 1 (

,)2 + « 2)2

Basing on (6), we obtain the following connections:

(7)

{n = 0, 1, ...),

(n = 1 ,2 ,...) ,

(n = 0, 1, ...),

(w = 1, 2, ...),

(n = 0, 1, ...),

(n = 0, 1, ...).

^n,l + Л ,+ 1.2 —0 (n = 0, 1, ...) for *2 = a,

^n,l

0 II <N

+ (n = 1, 2, ...) for *2 =

a

^n,2

Bn+

= 0 (n = 0, 1, ...) for *2 = - a

Bn,

+ #n,l = 0 (n = 1, 2, •••) for *2 = a, L„,l = Cn+ 1,2 (n = 0, 1, ...) for *2 = я,

^«,2 = Cn+ 1,1 (n = 0, 1, •••) for *2 =

a L

1. 77ie series

<8> f DM'4-.

+'4»,2). L В*ДВ,,,1 + В„,2|

n=1 n= 1

0 = 1 ,2 ; к = 0, 1, 2) are uniformly convergent in the set D x JR.

The series

(9) t £>*([ ( - l ) ”( Q . - C „ +1.2)] (i = l , 2 ; * = 0 ,1 ,2 )

n — 2

are uniformly convergent in the set Db x [ — a, a], where Db = \X: 0 ^ x

^ b,

|x2| ^ a] , h being a positive constant.

P ro o f. For the expressions Dkx.(An<

+ A n2), Dk.(Bnl + B„2) {k = 1, 2;

i = 1, 2; n = 1, 2, ...) we obtain by (6) the estimations

(10) \Ок.(ЛпЛ+ А пЛ)\ ^ M (2« — l ) - 2 , \Dk.(Бп>1 + Bn>2)\ ^ M ( 2 n - 1 ) ~ 2, (i, к =

2; n =

2, ...) for (X , y ^ e D x R , where M is a convenient posi­

tive constant.

It follows from (6) that

(11) К ( - 1)"(C„, - c n+, ,2)| M , (n - i r 2

(i = 1, 2; * = 0, 1, 2; n = 2, 3, ...) for ( X , b e i n g a convenient positive constant.

Applying the mean value theorem to the functions А пЛ+ А пЛ in the interval [ — a, x2] and to the functions Bn>

+ B

in the interval [x 2, я] we obtain by (7) the estimations

( 12)

\АпЛ+ А пЛ\ ^ M

(a + x

)(2n — l ) ~ 2, \В

+ ВпЛ\ < M

( a - x

) ( 2 n - l

(n = 1,2, ...) for (X, y x) e D x R , where M

is a convenient positive constant.

It follows from (10), (11) and (12) that series (8) are uniformly convergent in the set D x R and series (9) are uniformly convergent in the set Db x l ~ a , a] .

3. We shall prove that under some conditions imposed on functions f (i = 1, 2) the function

(13) where

u(X) = Ul (X) + u

(X),

щ ( Х ) ^ ( -

У

2

n + Q /% O

f ( y i ) Gi ( X, y l) dy l

— 00

(i = 1,2)

and G, (i = 1, 2) are given by formulas (5), (6) is the solution of equation (1) in the set D satisfying the boundary conditions (2).

4. We examine in this chapter the properties of the functions щ 0 = 1,2).

Let

л (ЛГ, yx) = £ ( A ^ + A„,2), n= 1

В ( Х , ^ ) = £ (B„,1+B„.2).

n= 1

Let us write the functions щ (i = 1, 2) in the form (14) щ(Х) = ми {Х) + щг

(Х) (i = 1, 2), where

OO

— 00 00

«

W =

J Z iO h M p r, У1МУ1, 1 *

« 2 , i W = - J f 2 ( y i ) B 0,2(X ’ y j d y i ’

- 00

ao

«2.2

— CO

Let

00

[«■,!(*)]? = I f l (yi)Dk

A

_l ( X , y l) dyl ,

—

t » i , ! W Ï = 7 [ f A y , ) O kxlA ( X , y l )dyl ,

— 00 00

— 00

oo

[«

W

= f j / л у а ^ щ х , y i )dy ,

— 00

O' = 1, 2; k = 0, 1, 2) and [и*,,(ВД° = uM(X) (i, k, l = 1, 2).

Now we shall prove the following

L emma 2. Let the functions f (i = 1, 2) he measurable in R and let 00 1

1

\fi(yi)\dyi

1° the integrals [м1Л(2f)]f (/с = 0, 1, 2; / = 1,2) are locally uniformly

convergent at every point X e R x ( — a , a ] \

2° the integrals [

2>1 (X)]f (к = 0, 1, 2; i = 1, 2) are locally uniformly convergent at every point X e R x \ _ — a,a);

3° г/ге integrals [ai)2(X)]f (i, / = 1, 2; к = 0, 1,2) are /oca//y uniformly convergent at every point X e D ;

4° the functions щ (i = 1, 2) satisfy equation (1) m the set D;

5° Mi,2 (X) -► 0 w/iea X -* (x?, — a), X e D, u

(X) -> 0 when X -> (x?, a), X e D , м1(Х )-^ 0 when X -►(х$), a), X e D ,

2(X )->0 when X ->(x°, —a), X e D .

P ro o f. Since

Щ А

Л {Х> yi)\ < ^ ( a + x 2) I ^ B o .2№ J>i)l<JV(a-X2)

fc 1 for Х е Я х ( - а , a],

~k-1 for X e R x [ — a, a)

{к =

2; i =

2), where iV is a convenient positive constant, we have ([«1,1 W J il < 2Vi(л-f-ЛГ2)~*~1 f lfi(yi)[dyi for X e R x ( - a , a], (15)

|[«2,i W ]?l < N ^ a -Х г) k 1 J I/2CF

) I ^

for X e R x [ - a , a ) (A = 0, 1, 2; i = l,2 ) ; being a convenient positive constant.

By inequalities (10) and (12) we obtain

(16) |[% 2f f l ] f K l V 2 £ ( 2 П - 1 Г 2 J If A y ^ d y , n= 1

(A = 0, 1 .2 ; i = l,2 ) for X e D , where N

is a convenient positive constant.

By (15) and (16) we obtain l°-3° of Lemma 2. Since for an arbitrary fixed point y}e R the functions A ni, Bni given by formulas (6) satisfy equation (1) in the set D, by Lemma 1 and l°-3° of Lemma 2 we obtain 4° of Lemma 2.

In view of (7), Lemma 1 and l°-3° of Lemma 2 we obtain 5° of Lemma 2.

L emma 3. Let the function L be measurable in the set R and continuous at 00

the point x ° e R . Let J \L{y\)\dyi < oo. Then the function

L(X) = x2

^(ЗЧ Ш У !-*!) + (x 2) ] dy 1

is defined for x x > 0, |x2| < oo and convergent to L(x®) when X ->(x°, 0).

The lemma above is a certain modification of the theorem from [4]

(p. 263, vol. I) and its proof is similar to that of the theorem.

It follows from (14) and Lemmas 2, 3 the following

T heorem 1. Let the functions f { y x) (/ = 1, 2) be continuous in R. Let

00

1 \ f ( y i ) \ d y l <

(i = 1,2).

- 00

Then the function и defined by formula (13) is the solution of the boundary problem (1), (2) in the set D.

5. Let X e D , , Y e D , = {Y: y, > 0, \y2\ ^ a}.

Let us denote by X = ( — x l5 x2) the symmetric image of the point X with respect to the axis y 2. Then

Y) = v / ^ . + x , ) 2+(</„.,)2 (i = 1, 2; n = 0, 1 ,...).

Let us consider the function ( n )

i — 1

where

Vi(X) = (~

T j e M L G d X . y ^ - G d X . y ^ d y i (< = 1,2), 0

v

( X ) = - ^ - j g

{y

)G

{X, y

)dy

and G, (i = 1,«2, 3) are given by formulas (5), (6).

Now we shall prove that under certain assumptions concerning the functions g, (/' = 1, 2, 3) the function v defined by formula (17) is the solution of equation (1) in the set D, with the boundary conditions (3), (4).

We shall prove the following

L emma 4. Let the functions gt (i = 1, 2) be continuous in the interval 00

[0, oo) and let j \gi (yi)\dyl < oo (i = 1, 2). Let (x?, x

) e D , . Then

— 00

1° the functions (i = 1, 2) satisfy equation (1) in the set D,, 2° ц ( Х ) - * 0 (/ = 1, 2) when X - * ( 0 ,

J), X e D ,

3° Vi(X)-*

i(xf) when X ->(x?, ( - l ) 'a ) , X e D , 0 = 1, 2), 4° i?,.(20->0 when X -+(xj, ( - l)i + 1 a), X e D , 0 = 1,2).

P ro o f. Observe that when we set

fiiy i) 9i(y i) for y

^ 0

-

Л - У

) f o r y ! < 0

the functions v( (i = 1, 2) take the form

Vi(X) =

2 к f i(yi )Gi ( X , y

) dy

(i = 1,2), i.e. the function v{ has the form of щ (i = 1, 2).

Since Gi(X, y ^ — GiiX, yj) = 0 for x t = 0 and by Theorem 1, we obtain l°-4° of Lemma 4.

Now we shall deal with the function v3. Let C ( x , y 2) = £ ( - i r [ c „ , 1- c „ +1,2;|.

n= 2

Let us write the function v

in the form

(18) v 3 (X) = I(X) + I 0 jl( X ) - I U 2 ( X ) - I u l ( X H I 2 , 2 ( n where

-1

l j ,k W = ~ I ^(У

⁾ Cj.k (X, У

) dy2

(/, k ) e Z = {(0, 1), (1, 2), (1,1), (2, 2)}

1 { X ) = — - 1 g

(y

) C ( X , y

) dy2.

к Let

a

U

A V ]f = ^ ( j , k ) e Z ,

m m = - 1

71 a

д Л у

) ^ р х. С( Х, y

) dy

J

— a

(/ = 1 ,2 ; p = 1, 2) and [ /M (*)]? = / M (*), [ / (X)]P = / (X) ((/, k) eZ, i = 1, 2).

Now we shall prove the following

L emma 5. Let the function g

be measurable in the interval [ — a, a] and

a

let j \g

(y

)\dy

< 00.

— a

Then

Г the integrals [ /j>k(2f)]f, { j , k ) e Z ,

= 1 ,2 ; p = 0 ,1 ,2 , arc locally

uniformly convergent at every point l e ( 0 , 00) x [ — a, à];

2° the integrals [/(2Q ]f (/ = 1 ,2 ; p = 0, 1, 2) are locally uniformly con­

vergent at every point X e [0 , o o )x [ —a, a];

3° the function v

(X) satisfies equation (1) in the set Dt ; 4° u3(x)->0 when X ( —1)‘a) (/ = 1, 2), x? > 0, X e D j.

P ro o f. Since

Щ С и (Х, y2) K for Z e (0 , o o ) x [ - a , a], y2e [ - a , a]

(p = 0, 1, 2; / = 1 ,2 ; (j, k)eZ), where is a convenient positive constant, we have

(19) |[/j,n(X)]f| < h

{x1)~p~ l ] \g

{y

)\dy

for X e (0 , o o ) x [ - a , a ]

— a

(p = 0, 1 ,2 ; / = 1 ,2 ; (j, k) eZ) and h

being a convenient positive constant.

By inequalities (11) we obtain

co a

(20) l[f(20]fKft3 I ( « - 1 Г 2 j \вЛУг)\Луг

1 n= 2 - a

for 2 fe[0 , b] x [ — a, a] (p = 0, 1, 2; / = 1, 2), where h

is the convenient positive constant. In view of (19) and (20) we have 1°, 2° of Lemma 5. Since for an arbitrary fixed point y

e( — a, a) the functions Cni (/ = 1, 2;

n = 0, 1, ...) as functions of the point X satisfy equation (1) in the set Dlt thus by (18), Lemma 1 and 1°, 2° of Lemma 5 we obtain 3° of Lemma 5. In view of (7), Lemma 1 and 1°, 2° of Lemma 5 we obtain 4° of Lemma 5.

L

6. Let the function g

be measurable in the interval [ — a, a] and

a

continuous at the point x 2e( — a, a). Let J \g$(y

)\dy

< со. Then

—a

v

( X) - +g

(x°2) as X -» (0, x°2), X e D ,.

P ro o f. In order to prove Lemma 6 it is enough to show that (21) J 0. i ( X ) ^ g 3(x?) when A"->(0, x2), X e D t ,

(22) / M ( X ) ^ 0 when X ->(0, x°2), X e D t , {j,k)e !(1, 2), (1, 1), (2, 2)J, (23) /(2 f) — 0 when X - ( 0 , x % X e D , .

The function /„,, can be represented in the form

CO

/ o.i(x) = ~ J <h(y

) l ( x

+ (y

- x

)

Y

dy2,

— 00

where g

(y2) = дъ (y2) for y

e [ - a , a], g

{y2) = 0 for y

e R \ [ - a , a]. In

view of Lemma 3 we obtain (21).

The proof of (22) is based on the inequalities

a

< h

x

(x

- a

j \g

(y

)\dy2,

— a a

\Ii,i(x ) \ ^ h

x

(x

+ a

{ \g

{y

)\dy2,

— a a

\ h ,

(x )\ J \g3M l dy

— a

for 2 fe D l5 where h

is a convenient positive constant.

By Lemma 1 and by 2° of Lemma 2 we obtain

a

lim /(X ) = — 1 j g

(y2) lim C{X, y

)dy

= 0, X e D l5

X - ( 0 ,x ® ) K

J

z - a 1

which proves (22).

As the consequence of Lemmas 4-6 we get the following

T

2. Let the functions ^ ( y j (i = 1, 2) he continuous in the interval (0, oo). Let the function дз(у2) be continuous in the interval ( — a, a).

со a

Let J ЫУх)МУ1 < oo O'= 1,2), J \дз(у

)\(

у

< oo. Then the function v

0 —a

defined by formula (17) is the solution of equation (1) in the set Dj with the boundary conditions (3), (4).

References

[1] P. L. B u tz e r , W. K o lb e and R. J. N e s s e l, Approximation by Functions Harmonic in a Strip, Arch. Rat. Mech. Anal. 44.5 (1972), 329-336.

[2] G. F. D. D u ff and D. N a y lo r , Differential Equations of Applied Mathematics, John Wiley & Sons, Inc., New York.

[3] V. I. G o r b a iC u k and P. V. Z a d a r e i, Properties o f the solutions of the Dirichlet problem for the disc and strip, Math. Collection (Russian), Izdat. “Naukova Dumka”, Kiev 1976,

64-68.

[4] M. K r z y z a n s k i, Partial Differential Equations o f Second Order, vol I, Warszawa 1971.

[5] D. V. W id d e r , Functions Harmonic on a Stripe, Proc. Amer. Math. Soc. 12 (1961), 67-72.

[6] —, Fourier Cosine Transforms Whose Real Parts Are Non-negative in a Strip, ibidem 16

(1965), 1246-1252.