ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I. COMMENTATIONES MATHEMATICAE XXI (1979) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I. PRACE MATEMATYCZNE XXI (1979)
M. Filar (Krakôw)
On a certain boundary problem for the equation
A 2 u - C * u = f ( X )
1. In this paper we construct the solution of the equation (1) A2u ( X ) - C * u ( X ) = f ( X ) ,
where X = (xl 5x 2) and C is a positive constant, for the half-plane D = {X: IxJ < o o,x2 > 0}
with the Riquier conditions:
м(х!,0) = / 1(x1), Au{xx,0) = f 2{xx).
The functions f (i = 1 , 2 ) are defined for x x edD, where ÔD denotes the boundary of D. f is a given function on the set D.
2. We give now some formulae and theorems which will be needed later. Let X and У be two different points in 2-dimensional Euclidean space E2. Let
r = Х У = l ^ - y t f + i x . - y j y i 2.
As we know [5], the fundamental solution of the equation
(la) A2u ( X ) - C * u ( X ) = 0
is the function
(2) U(r) = nY0(Cr)+2K0(Cr),
where Y0(z) is the Bessel function of order zero of the second kind and K 0{z) is the Mac Donald function of order zero [3]. Let X e D , Y e D
= DkjôD. Let us denote by X x = (xl5 — x2) the symmetric image of the point X with respect to the axis y2 = 0. Further let us write
rx = X x У = [(xi — y i )2 + (x2 + y2)2] 1/2, Q = [ ( X i - y ^ + x i ] 172.
The following theorems are true (see [1]).
Theorem 1. The function
G( X, Y) = U ( r ) - U ( ri), (3)
where U (r) is given by (2), is a Green function with the pole X for equation (la) and for D with the boundary conditions
G ( X , Y ) = 0, AyG ( X , Y ) = 0 for YedD.
Theorem 2. Let f ( i = 1,2) be two functions measurable in ÔD and
00
continuous at the point x^^edD. Let j \fiiyi)\dyi *< oo (i = 1,2). Then the
— 00 function
00
(4) щ( Х) = (4nC)~l x 2 J [ f i ( y i ) C2N 1(Q)+f2(yl) N 2{eï]dyl ,
— 00 where
Nde) = е ^ И -^ я г а о д + г х п о ? )] O’ = i ,2)
is the solution o f equation (la) in the domain D and satisfies the following boundary conditions:
lim Mi(A') = / 1(3c1), lim Au^(X) = f 2{xx) as X- +( xl t O+).
3. We shall now examine some properties of the function U (r) given by formula (2) and prove some lemmas justifying differentiation of the integral of the form V{X) = J / (Y)U(r)dY.
D
If we use the formulae [3]
~ [ z - " K „ ( z ) ] = - z - K . + 1(z), 4 L [ z - " X ( z ) ] = - z - " y . + 1(z),
dz dz
the identities
Ax Y0 (Cr) + C2 Y0 (Cr) = 0, Ax K 0( Cr ) - C2K 0(Cr) = 0,
and asymptotic properties of the Bessel functions Yn and Mac Donald functions K n, the following will easily be obtained:
(5) U(r) — о (r2 y) ) > D^Xi U(r) = o(rA v), Dxi DX2 U (f) == 0 (1), D2L^X, U(r) = o(r~y)
(i := u : о V V
1) as r - 0 , (6) Ax U (r) = o (r_v), DXl^ x \U(r) == 0 ( r - ‘)
(i = l , :ГЧ1 о V r- V 1) as r->0 , (7) Dli D'x2 U (r) = 0 (1) as r-> oo (a, = o, 1, 2 ,3 ,4 ; a +«a. V/
- L j V(r) = C3 [nYt (Cr) —2X, (Cr)] = —4C2 r ~ l +h(r), dr
where h(r) = 0 (1) as r -+0 (r-* + co).
(8)
On a boundary problem 285
We shall prove the following
Lemma 1. I f the function f is measurable and bounded in D and S\ f ( Y) \ d Y < oo, then the integrals
D
K , m = j f (Y)£T(l U (r)dY (a, fi = 0 , 1, 2; a + fi ^ 2),
D
Щ(Х) = $ f ( Y ) D KAx U(r)dY ( / = 1 , 2 )
D
are uniformly convergent at every point X 0 eD.
P ro o f. It follows from (5) and (7) that there exists a number M > 0 such that
\D*X1 Di2 U(r)\ ^ M for r > 0 (a, p = 0, 1).
Therefore
\Kp(Xy sS M S \ f ( Y) \ dY for X e D (a, = 0, 1)
D
and the common majorant of integrals VxfS(X) (a, f = 0, 1) is the convergent integral M § \ f (Y)\dY.
D
We shall now prove that the integral Wx (X) is uniformly convergent at the point X 0 e D. The proof for the integrals W2(X), V20(X), and V02{X) is similar. It follows from (6) and (7) that there exist numbers R > 0, M, > 0 (i = 1, 2) such that
(9) \DXiAx U{r)\ ^ M xr ~l
M 2
for 0 < r < 4R, for r ^ 2R.
Let K ( X 0,3R) denote a circle with the centre X 0 e D and radius 3R.
We shall write Щ (X) in the form:
Щ( Х) = W' (X)+ W?(X), where
Wi ' ( X ) = s f ( Y ) D x l Ax U(r)dY,
D n K ( X 0 , 3 R )
W? ( X ) = f f (Y)DXI Ax U(r)dY.
D\K(X0 ,3 R )
For X e K ( X 0, R) and Y e K { X 0, 3R) we have r = X Y ^ X X 0 + Y X 0 < R + + 3R = 4R. If X e K ( X 0, R ) and Ye D\ K ( X 0, 3R), then r = X Y = X 0 Y -
— X 0X ^ 2R. From these results and from formula (9) we obtain l^ i W I ^ |И ? (* )| + |И ?(Х)|
S M , J I f ( Y ) \ r ~ 4 Y + M 2 I \ f [ Y ) \ d Y
D n K ( X 0 , $ R ) D\K(X q, 3 R )
for X e K ( X 0, R) (cf. [4]).
It follows from the above inequalities that the integral Wl {X) is uni
formly convergent at the point X 0 e D.
We get as corollaries of Lemma 1 the following
Corollary 1. I f a function f satisfies the assumptions o f Lemma 1, then the function
(10) V( X) = l f ( Y ) U ( r ) d Y
D
is of class C2 in D and
ITxl Di2 V ( X ) = Vap(X) for X e D (a ,0 = 0 ,1 ,2 ; a + 0 ^ 2).
Corollary 2. I f a function f satisfies the assumptions of Lemma 1, then the function
(11) W( X) = § f (Y)Ax U(r)dY
D
is of class Cl in D and
DXiW( X) = Wf X) for X e D (i = 1,2).
We now prove
Lemma 2. I f a function f is bounded and o f class C 1 in D and f I / (Y)\dY < oo, then the function W (X) defined by formula (11) is o f class
D
C2 in D.
P ro o f. Let the circle K ( X 0, 3 R ) <z D and conditions (9) hold true. We now present the function W (X) given by formula (11) in the form
(11a) W( X) = L{X) + H( X) ,
where
Ц Х ) = I f ( Y ) d x U(r)dY, H ( X ) = f f ( Y ) A x U{r)dY.
K (Y 0,3 R ) D\K{X0 , 3 R )
Let X e K ( X 0, R). Then by (7) the function H(X) is of class C2 in K ( X 0, R) and its derivatives up to the order two may be found by dif
ferentiation under the sign of the integral. Taking into consideration the above properties and the fact that the function U(r) as a function of the point X (X Ф У) satisfies equation (la) we have
(12) dH (X ) = C4 J f ( Y ) U ( r ) d Y for X e K ( X 0, R).
D\K(X0 , 3 R )
Using the formula
DXi Ax U (r) = - Dyi Ax U (r) (i = 1,2) and Corollary 1, we get
DXiL ( X ) = - f f ( Y ) D yiAx U(r)dY (i = 1,2).
K (X 0,3R )
On a boundary problem 287
From the formula for integration by parts [2] we obtain (13)
Dx, L ( X) = f Ax U(r)D„f(Y)dY+ J f (Y)dx U(r) cos (nr , yt)dSY
K(x0,m ew„,iR)
for X e K ( X 0, R ) ( i — 1,2), where nY denotes the inward normal to d K ( X 0,3R). Formulae (13), Lemma 1, and Corollary 2 imply that the function L( X) is of class C2 in K ( X 0,R). Thus the function W( X) is of class C2 in K ( X 0, R) and hence also at the point X 0.
Lemma 3. I f the function f satisfies the assumptions o f Lemma 2, then (14) lim A L ( X 0) = - 8nC2f ( X 0) as 0, X 0 e D.
P ro o f. By Lemma 2 and (13) we have D l . UX ) = f D , J ( Y ) D x, Ax U(r)dY+
K(X0,iR)
+ f f ( Y ) D XiAx U(r) cos (nY, yJdSy êK{X0,3R)
2
for XgK ( X 0, R) (i = 1,2). For D2fL(X 0), according to the formulae i = 1
DXiAx U(r) = - DyiAx U(r) ( i = 1,2), we obtain
AL(X0) = i D2L ( X„ o) = + i= 1
where
B i(*o) = J I DyJ ( Y ) D „ d x U W x = xodY, K(X0,iR) i= 1
— I Ax U(r)\x =XodSY.
SK(X0,3R) OnY
The integral B l ( X 0) is an integral of the type W* ( X0) and can be made arbitrarily small by selecting the sufficiently small radius 3R. It is enough to show that
(15) lim B2( X0) = - 8j z C2f ( X 0) as Д - 0 .
Since on the boundary d K ( X 0, 3 R ) of the circle K ( X 0,3R) we have
ô d
—— Ax U (г)|х=х0,УегК(Х0,ЗК) = — — Ax U (г)|г=зл,
cnY dr
we get by (8)
dU(r) B2(X o )= f f ( Y )
гк{х„, 3R) dr dSx
X = X o
— AC2 +h(r)
= дК(Х0, ЗЛ)J / ( y )
Applying the mean value theorem to the last integral, we obtain
— AC2
dSv .
r = 3 R
B2(X o ) = 2n3Rf{Q)
3 R + h(3R) where Q e d K { X 0,3R).
Then, by the continuity of / for X = X 0, we get (15).
Lemma 4. I f a function f satisfies the assumptions of Lemma 2, then the function — (SnC2)~1 V (X), where V (X) is given by formula (10), satisfies equation (1) in D.
P ro o f. From Corollary 1 and Lemma 2 it follows, by formula (11a), A2 [_aV{X o)] = a AW( X0)
= aAL(X0) + a AH( X0) ± C 4 J af(Y)U(r)\x=XodY,
K (Xq, 3 R )
where K ( X 0, 3R) cz D and a = —(SnC2)- 1 . By (12) we get
A2 [ aV( Xo)] = aAL ( X0)+aC4 V ( X 0) — C4a j f ( Y)U (r)\x=XodY.
K ( Xq, 3 R )
Since
lim a f f ( Y) U{r) \ x=Xod Y = 0 as R-+0,
K ( X 0, 3 R )
we have by (14)
A2 [ a V( X0])] = f ( X 0) + aC4 V ( X 0) for every X 0 e D .
Lemma 5. Let f be function measurable and bounded in D. Let {I f ( Y ) \ d Y < oo. Then the function A{X) = j / ( Y) U( r 1)dY is of class C4
D D
in D and satisfies equation (la) in this set.
P roof. We shall prove that the integrals
A ^ ( X ) = i f ( Y ) D l , D i 2 U(ri) dY («,0 = 0, 1, 2, 3, 4; ce+0 « 4) n
are uniformly convergent at every point X 0 e D. Let K { X 0, R) <^ D. For X e K ( X 0, R) and Y e D we have r1 ^ x 2 ^ S > 0, where <5 is a positive constant. From formulae (7) we see that the functions D*x tD!fc2U(rl) (a, p
= 0, 1,2, 3 ,4 ; ol + P ^ 4) are bounded for r1 ^ <5. From these results we can obtain the inequalities
K j W I H M , f $ \ f ( Y ) \ d Y . (a, 0 = 0 , 1 , 2 , 3,4; a + 0 « 4)
D
On a boundary problem 289
for X e К ( X0, R), where Млр are positive constants. It follows from the above inequalities that integrals Aap(X) (a, p = 0, 1,2, 3, 4; ot + f ^ 4) are uniformly convergent at the point X 0 eD. Thus the function A(X) is of class C4 in D and DX1 D^x2 A( X) = Aap(X) for X e D (a, jS = 0, 1,2, 3, 4;
a + /i ^ 4). Taking into consideration the above properties and the fact that the function U (rx) as a function of the point X satisfies equation (la) in D, we have
A2A { X ) - CaA( X) = \ f{Y)\_A1x U{rx) - C * U { r x) ' ]dY= 0 for X e D .
D
4. As an immediate corollary of Theorem 1 and Lemmas 4, 5 we get Theorem 3. I f a function f is bounded and o f class C 1 in D and J \ f (Y)\dY < oo, then the function
D
(16) u2(X) = - ( 8nC2) - ' J /( Y ) G ( X , Y)dY
D
is the solution of equation (1) in the domain D with the boundary conditions:
lim u2(A) = 0, \im Au2(X) = 0 as X -* X e d D , X e D . As a consequence of Theorems 2, 3 we get the following
Theorem 4. I f functions f (i = 1,2) satisfy the assumptions o f Theorem 2 and a function f satisfies the assumptions o f Theorem 3, then the Junction
u(X) = u1( X) +u2(X),
where ux (X), u2 (A) are defined by formulas (4) and (16), respectively, is the solution of equation (1) in the domain D with the boundary conditions :
lim u(X) = (xx), \\m Au{X) = f 2{xx) as Х^>(х1, 0+) , X e D .
References
[1] M. F ila r, On a certain boundary problem for the equation A2 u — C*u = 0, Comm. Math.
16 (1972), p. 83-90.
[2] M. K r z y z a n s k i, Rôwnania rôzniczkowe czqstkowe rzçdu drugiego, cz. I, Warszawa 1957 (in Polish).
[3] N. N. L e b ie d ie w , Funkcje specjalne i ich zastosowania, Warszawa 1957 (in Polish).
[4] H. M a r c in k o w sk a , Wstçp do teorii rôwnah rôzniczkowych, Warszawa 1972, p. (89 (in Polish).
[5] J. M u sia le k , Construction o f the fundamental solution for the equation A2 u + ku = i), Comm. Math. 9 (1965), p. 213-236.
