ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEQO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE X X (1978)
Jàn Gorowski (Krakôw)
O n the Diriclilet problem îor interior and for exterior of an elliptic domain
1. In the present paper we shall give a solution of the Diriehlte problem for the domain
( а?2 у г )
— +-j^<l, 0<4<aJ
and
_ [ x 2 y 2 I
Gz = CGX = |(®,y): — > 1, 0 < b < a j.
We shall construct
1° the function и such th at
(1) Au(x, y) = 0 for {x, y) e Gx
and
(2) ' и(я,У) = f ( x , y ) for (x , y ) e d G : 2° the function w such th at
(3) Aw(x, y) — 0 for (a?, y) e Gz,
(4) У) = 9(x, У) for ( x, y) e dGx
and w(x, y) is bounded in the domain Gz.
The given functions f, g are continuous on the boundary dGx and f{aco&<p, & sin99) is a function with of the bounded variation with respect
a e [ 0 , a 0] ,
<p e [0, 2n], where e > 0, a = ccosha0, b = csinha0.
to the variable <p (<p e [0, 2n]).
2. Let
x = ecoshacos<
y = csinliasinç (5)
338 J. Grôrowski
By the transform (5) we get
(6) AU =
c2(sinh2a + sill2*??) (■^ao U + U) — 0,
In the sequel we shall use method of separation of variables. We assume th a t
u = U M - U M , where U2e (72(0, 2n) is a periodic function.
Consequently we get the system of the equations (7) Ü Ï M + W ^ a ) = 0, U2 (<p)-AU2(<p) = 0 and we obtain À = —n 2 (n = 0 , 1 , 2 , . . . ) . Hence
(8) U " ( a ) - n 4 J x(a) = 0, U2 (<p) + n 2U2(<p) = 0. The functions
Ui (a) = Л 0+ В 0а (n = 0 ) ,
Z7“(a) = A wcoshw a+B wsinhwa (n — 1 ,2 , ...), U2{(p) = А'по,о%щ +B'nmin(p (n = 0 , 1 , 2 , . . . ) .
A 0, A'0, A n, A'n, B 0, B'0, B n, B'n being constant they are the solutions of system (8).
Consequently we get U ° = C 0+Doa (n = 0),
Un = СпеовтрсовЬпа+Впсовп<рвтЪ.па +
Л-Епsinmpcoshna + E ns in ^ s in h ^ a (n = 1 ,2, ...).
We assume th a t
(9) (grad U)2 < oo.
We have
(10) (grad U f =
c2(sinh2a + sin2ç>) and by (9) and (10) we obtain
Д) = 0, В n = E n = 0.
Hence the solution of equation (6) has the form
{ { B ^ f + ^ U f )
(11) U(a, <p) OlQ
~2
v i /coshna sinlma \
> — :---
апъовп(р-\-
-г-:--- ônsinwœ Z» = j1 \coshwa ' 0 n r sinhwa0V IDirichlet problem 339 Let
(12)
and
1 “n
an —— I f(t)cosntdt, n — 0 , 1 , 2 , . . . , n J0
1 2я
bn = — f(t )si nntdt, n = 1,2, . . . ,
7C J
«о / cosh wa p . v ( « , ? ) = T + 2
П = 1
sinhwa . \ bnsmrup ,
sinhwa0 /
For the sequence y)} the asumptions of the Harnack theorem are fulfilled and consequently the function
an vh / coshna sinhna 7 . \
U{a,q>) = — + } ---- :--- ano,o%ncp + — ---
2 \coshwa0 sinhwa0 /
H = 1 is a solution of problem (1), (2).
By the Ham ack theorem we get
2тг o°
<l3> Р(“’^ = ^ / /(()[1+2Л
\H / coslmacoshwa0c o s nt C O S Щ)
smhwa . . \1 , -1--- smntsmmp) \dt
sinhna0 / J
and finally we get the following
T h e o r e m 1. The function defined by formula (13) is a solution of problem (1), (2).
3. Now we shall construct the solution of problem (3), (4). By the transform
x — ccoshacosç?, a e [a0,oo), y = csinhasinç?, <p e [0, 2л], (14)
we get equation (6).
Similarly as in the foregoing p art we get W°(a, q>) = A 0+ B 0a {n = 0),
wn(a, cp) = A ne~naeosncp + B nenaco$n<p +Cnenasinncp +Dne~nasinncp (n = 1, 2, ...)•
Assuming th a t the function w is bounded we obtain B o = 0, B n = = 0.
340 J. Grôrowski
Hence the solution of equation (6) for the domain 0 2 has the form (15)
Let
0 V—1 ô
w(a,<P) : p ^ ( cnCOS^ + dnsinw<p).
2rc
cn
= — I
g(t)co$ntdtTZ J
0 1 27T
= — I
g(t)sinntdt TC J 0( n = 0, 1,2, . . . ) ,
(w = 1» 2 » •••)•
I t follows from the uniform convergence of the series (15) in the set (a0, oo) x [0, 2tc] th at
2tc oo
w(a,<p) = -^—J 9(t)j^l + 2 en(a°~a)cosn(t —ç?)jd£.
0 «-1
By ([2], p. 50)]Jwe obtain (16) w (a , 99)
2jt
i / * ( , )
sinh(a — a0)
cosh(a —a0) —cos(<- dt
1 ? 1 — (ea°~a)2
— «(Л ---dt.
2tc J yv ' l - 2eao~“c o s(* -9>) + (ea«-a)2
By ([!]? P- 259) the function defined by formula (16) satisfies the boun
dary condition (4) and finally we get the following
Theorem 2. The function defined by formula (16) is a solution of problem (3), (4).
R e f e r e n c e s
[1] M. K r z y z a n s k i, Bownania rôzniczkowe czqstkowe rzçdu drugiego, tom I, War
szawa 1957.
[2] I. M. R y z y k i I. S. G r a d s z te jn , Tablice calek sum szeregow i iloczynow, Warszawa 1964.
