A N N A LE S SO C IE TA T IS M ATH EM A T IC AE P O LO N AE Series I : COM M ENTATIONES M ATH EM A T IC AE X V I I I (1975) R O C ZN IK I P O L SK IE G O T O W A R Z Y S T W A M ATEM ATY CZN EG O
Séria I : P R A C E M ATE M A TY C ZN E X V I I I (1975)
Feliks Bauanski and Etjgenitjsz Wachnicki (Krakôw) Orlicz space and Diriclilet problem
In the paper we shall give the concept of the generalized Dirichlet problem in Orlicz space for the simple connected domain.
1. Let M(u) denote an arbitrary JV-f unction. Let L*M denote Orlicz space of the functions defined in the interval [a, b] and let (|• ||M denote the norm in L*M. Further let L*M{A2) denote Orlicz space if M(u) satisfies the condition M (2u) < JeM (и), и > 0, Tc being a positive number.
For LM(A2) t
ь
ll/» -/ll« -> 0 о / M(\f„ (t)-f(t)\)dt^0
a
if n-> oo ([1]).
2. Let 8 — {(x , y ): x — <p{t), y = y(t), te [a, &]} denote the boundary of class G1 of the simple bounded connected domain D. Let { 8n} denote an arbitrary sequence of the closed curves of class C1 for which x = <pn(t), y — y)n{t), te [а, Ъ] and
8n c B(cp{a) = 99(b), y>(a) = y{b), cpn{a) = (pn(b), y>n{a) = y n{b)).
Definition. The sequence {$n} satisfies condition (8) if q>n{t)-><p{t), y>n{t)-^^{t) uniformly in the interval [a, 6].
Let f ( X ) , X = { œ , y ) , denote an arbitrary function defined on 8.
Let u( X) be a function defined in D. Let un(X) — и (X) for X c 8n.
Definition. un(X)->f(X) in Orlicz norm if Un(t) = u(q>n(t), ipn{t)) converge to the function F(t) = f(<p(t), <p{t)) in Orlicz norm.
3. The function u(X) is called a generalized solution of the Dirichlet problem for the bounded domain D if u{X) is a harmonic function in D and for arbitrary sequence 8 n c z D satisfying condition (8) un( X ) — > f ( X )
in Orlicz norm. n~*°°
Let G(X, Y) denote the Green function for the domain D with a pole X . For a simple bounded connected domain D with boundary of class C1 such function G(X, Y) exists and is unique ([2]). 2
2 — R oczn ik i PTM — P ra ce M atem atyczn e X V III
174 F. Bararïski and E, Wachnicki
Let
(1) U
dG (X, Y) dnv doYl w-inward normal.
Let F{t)eL*M and
b
Jn{ F , t ) = j F { s ) K n{s,t)ds.
a
Lemma 1 ([3]). I f K n(s, t) is a sequence of the functions measurable in the square P = and there exists a constant A > О such that
b ь
f \Kn(s,t)\ds J \Kn( s , t ) \ d t < A
a a
for almost all t e [ a , b ] and se [a , 6], and if for every function F(s) e H, H being an every where dense set in L*M the condition
b
(2) lim jM(\.Jn( F , t ) — Jm( F , t ) \ ) d t = 0
те-* oo a
тег-* oo
is satisfied, then condition (2) holds for every F e L*M.
It is known that a set of continuons functions is everywhere dense in L*1(A2) and L*M{Af) is a complete space ([1]). From Lemma 1 follows Theorem 1. I f K n(s,t) satisfies the assumptions of Lemma 1 and for every function F e L*M( Af) continuous in [a, b~\
(3) \\J n{F, t)- F( ’ )\\M-^0 if n-*oo,
then condition (3) holds for every F e L*m(A2).
Let { 8n} denote an arbitrary sequence of the curves satisfying (S) condition. Let X e iSn', then
b
v n(t) = / F( s )X «( s , t)ds,
a
K n{s,t) 1 dG {X, Y) 2 к dnT
0 > ' ( * ) ] 2+ [ У М ] 2 F=(v>(s),y>(s))
-X=(<pn(t),v>n(t))
f lKn(s,t)lds
a
fbI-й-n(s, t)\dt
a
= 1 . Lemma 2.
Orlicz space and Dirichlet problem 175
P roof. From the Olejnik lemma ([2]) and a property of the Green function it follows that dG (X, Y)/dwF > 0 for X e D and Ye 8 and
j ь\Kn{s,t)\ds = dG{X, Y)
(ЛгТЪтг
Й(Т л — г2тс J
dG(X, Y)
dnv da i
If in formula (1) /(Y ) = 1, then ь
f !-£*(«, = 1.
a
Similarly,
ь
f \KAs ,t)\dt = 1.
a
Theorem 2. I / Fe L*M(Af), йеп Ш function u(X) defined by formula (1) is a solution of the generalized Dirichlet problem.
P roof. Since Fe L*M(Af), we have
f If ( Y ) \ d a < oo s
and integral (1) exists when X e D and
= 0 .
Since the domain D is regular with respect to the Dirichlet problem (2), thus for F(t) continuous in [a , b] JJn(t)->F(t) uniformly in the interval [a, b] and \\Un — F ||M->0 if n-+ oo.
From Theorem 1 it follows that \\Un — F\\M-+0 for every F e L*M(AZ).
4. In the particular case for the circle К : x 2 + y 2 < 1 from the Poisson formula
(4) Jr( F , t )
2tt
1 r 1 — r2
— F i d )--- --- de 2n J 1 — 2reos(0 — t ) + r we get
Theorem 3. I f FeI*M{A2), then Jr( F , t ) defined by formula (4) is a harmonic function in К and
as r->l.
176 F. Baranski and E. Wachnicki
References
[1] M. A. K r a sn o sie lsk iï, Ya. B. R u t ic k iï,. Convex functions and Orlicz spaces (in Russian), Moscow 1958.
[2] M. K r z y z a n s k i, Partial differential equations of second order, Vol. I, War
szawa 1972.
[3] W . O rlicz, Ein Satz über die Erweiterung von linearen Operational, Studia Math. 5 (1934), p. 127-140.
