Series I: COMMENTATIONES MATHEMATICAE XXX (1991) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE XXX (1991)
Maria Filar (Krakow)
Boundary value problems for the nonhomogeneous Helmholtz equation in a rectangular polyhedral angle
of the Euclidean w-space
1. Introduction. In [1] the construction and synthesis of solutions were given for certain boundary problems for the equation
(1) ' (A + c2)u(X) = 0
in the set
Q = {X: x t > 0, i — 1 , ..., n},
where X = (xl5 xn) is a point in the n-dimensional Euclidean space En (n> 2), A = £ " = xDli is the Laplace operator, c is a positive constant.
The purpose of this paper is the construction and synthesis of solutions of such boundary problems in Q for the nonhomogeneous Helmholtz equation
(2) (A+c2)u{X) = f(X ).
We seek the solutions of (2) which satisfy the homogeneous Dirichlet or Neumann conditions or the homogeneous boundary conditions of the third kind on the subset
S t = {X: x t = 0 , x k > 0 , k e { l , - , n } \ { ï } } (i = 1 ,. . . , n) of the boundary of Q.
For the construction of solutions we shall use convenient Green functions and the results of [1].
2. The operation о and its properties. Consider the sets N = { 1 , n}, W = {0, 1}, Wt = { 0 ,1 , u j, where vt > 0, ieN . Let A = W1x . . . x W n, В = W x ... x W (n times). We shall consider subsequences (vnt, . . . , v„J, heN , of the sequence {v1, ..., vn) and the subset A(nt , ..., nk) of A \B defined as follows:
n
А(п1У . . . , п к) = X Q(nl5 ..., nk)
i = 1
where
f M for nk}
for ie N \{ n l , ..., nk}, ie N
In virtue of the foregoing definitions,
(J
Л(пх, . . пк) = A\B, where we sum over all subsequences (rij, ..., nk), к e N of ( 1 ,..., n). Denote by {el , ..., en) the•standard basis of the space En. The elements a = (al9 an)e A will be identified with the vectors Y j= iaiei-
(XÏ1, . . . , x®"), where
f for II cT
Xi* = * for at = 1,
for at = vt, i e N , X (0....о) — -X"*
Let Y = (yl ,...,y„ )eE „ and let ra = \ Y - X a\ = © » i (У ~хТ)2)1/2, r(0,...,o) = г = 17 — X\. Let V{r) be a function defined for r > 0. We shall write for V(ra) (aeA). Let a e A (n 1, nk), k e N , and let
(3) I(Va) = (2 h)k J Va exp [h(vHl + ... + » J ] dvnx... dv„k,
R К
where h is a fixed negative number, and
R ; ...= {(»B1, • • • » 0 : v «i > 0 (* = 1, • • •, fe)} { k e N ) .
Definition 1. In the set of all functions Va, I{Va) ,a e A , a' eA \B , we define the operation о as follows:
1° VaoVa> = Va+a> for а, а', а + а 'еЛ ;
2° VaoI(Va>) = I(Va+tt') for a E B ,a !e A \B ,a + a’e A \B \ 3° I{Va)oI{Va) = I{Va+a) for a, a', a + a?eA\B.
The operation о is commutative and associative and has a neutral element V.
This operation is also distributive with respect to addition and multiplication by scalars.
3. Construction of Green’s functions for (l).and Q under certain boundary conditions. We shall now define some auxiliary functions which will be helpful in the formulation of mixed boundary problems for equations (1), (2) and the set Q.
Put Qt = {X: x { > 0}, S{ = {X : x t = 0}, ie N , Q = ü k j S Î и ... uS„+. As we know the fundamental solution of (1) has the form
(4) U(r) = (c r y vYv(cr),
v = (n — 2)/2, where Yv(z) is the Bessel function of the order v of the second kind [2]. Consider the functions
f G le. = Gle.(X, Y) = U + Ue., (5) < G2ei = G2ei(X, Y) = U - U ei,
U 3ei = G3ei(X, Y) = U + U et + I(UViei), where ie N , U is given by (4).
Let Ф = {1, 2, 3} x ... x {1, 2, 3} (n times) and let (p = {(p1, . ..,( p n) = (pl e1+ ...+(ряепеФ.
Consider the functions
(6) Gv = Gv(X, Y) = Gq>ieio ...o G <Pfn where G9jej (j e N ) are given by (5).
Let
(7) H № U J = (2hf f DxUaexp[h(vni + ... +Vnkÿ\dvni
“-1...
where a e A(nt , . . . , nk) (к e N), a denotes a sequence (at , ..., a j where af ^ 0, Dx stands for the derivative Dx\ ... Dxnn of order |a| = + ... + a„, 0 < |a| ^ 3.
Let N a = {ieN: af ф 0}. Now we state .
Lemma 1 [1]. 1° The integrals H(DxUa) given by (7) are locally uniformly convergent at every point
(X ,Y )e (Q t и St) x Qt [0, x (Qt u Sf)], ie N a;
2° D y ( l/a) = tf(D*x Ua) /or (X, [(fl, u SJ x f l j , ieiVe and /( ^ ) is defined by (3).
The following lemma deals with the properties of the functions (5).
Lemma 2 [1]. For к = 1, 2, 3 and i e N
1° Gkei is defined and of class C3 for (X, Y)eQ t x (Qt и St) [(Ц u St) x X * Y ;
2° Gke. satisfies equation (1) as a function of X , X Ф У, X e Q {, YeQ u St;
3° The following boundary conditions are satisfied:
(a) DXiGlei~*0, (b) G2et~+0,
(c) {DXi + h)G3ei = (DXi + h)Glei- 2 h U ei^ 0 as X - t X i e S ^ X e Q if Y is fixed in Qit Y Ф Х (ieN).
The following lemma concerns the properties of the functions (6).
Lemma 3 [1]. For (реФ
1° Gy is defined and of class C3 for
(X, Y)eQ x Q [Q x Q, (Q u Sf+) x (0 \S f+), {Û\St) x ( f iu S /)], X # У ;
2° Gy satisfies equation (1) as a function of XeQ with fixed Y eQ, Y ф X;
3° The following boundary conditions are satisfied:
(a) if (P i= l then DXiGy-+0, (b) if (pi = 2 then Gv ^> 0, (c) if <pt = 3 then (DXi + h)Gy-^0 as X ^> X te S t , XeQ, YeQ \S ? ,X Ф Y (î eN).
4. Formulation of the boundary problems for equation (2) in Q. We shall present now some explicit formulas for solution of some boundary problems for equation (2) in Q. In the construction of the solutions we will use the functions Gy defined by (6).
Let us consider the functions
(8) ur {X) = y , S f ( Y ) G r ( X , Y ) d Y n
where yn = n2~v~1c2v(T (v+ 1)0„)-1 and вп stands for the measure of the n-dimensional unit sphere; <p = (<pl t (рп)еФ; and f ( Y ) is a given function defined in Q.
We shall prove, under some assumptions on / (7), that uv is the solution of (2) in Q with the following homogeneous boundary conditions:
Г if (Pi = 1 then DXiUy(X)-+0, (9) < if q>i = 2 then Uy{X)-*0,
l i f <Pi = 3 then {DXi + h)Uy(X)~*0, as X - ^ X i E S ? , X eQ (ieJV).
5. The synthesis of the solution (8) of the problem (2), (9) in Q. We shall now investigate the properties of the function U(r) given by (4) and present some lemmas concerning the derivatives of the integral of the form
w(X) = y ^f(Y )U (r)d Y . Я
By the asymptotic properties of Bessel functions 7v(r) we get the formulas f U(r) = 0(r~s) for n = 2, 0 < s < 1 as r-*0,
(10) < U(r) = 0{r~n + 2) for n > 2 as r-> 0, lD XiU(r) = 0(r~n + 1) 0IeN) as r-> 0, (11) D*x U{r) = 0{ 1), 0 ^ |a| < 2, as r-> 00,
(12) Df U(r) =
where
F ( r ) e W - « )
for n = 2, 0 < s for n ^ 3 as r -
< 1 as
>0.
r —►o,
Lemma 4. I f the function f is defined, measurable and bounded in Q and
$a\f(Y)\dY < со, then
1° the integrals w(2f) and
Щ (X) = y J f ( Y ) D xlU(r)dY (i e N ) a
are locally uniformly convergent at every X e Q ;
2° w(X) is of class C1 in Q and Dx.w(X) = w fX) for X eQ (ieN).
P ro o f. By (10) and (11) there exist positive numbers R, M such that
|£/(r)| ^ Mr s for n = 2 when 0 < r ^ 4R,
\U(r)\ ^ Mr~n + 2 for n ^ 3 when 0 < r < 4R,
\DXiU(r)\ ^ M r_n + 1 (ieN) when 0 < r ^ 4R,
\DxU(r)| ^ M (|a| = 0, 1) when r ^ 2R.
Let K(X, 3R) be the ball of radius 3R and center at X e Q . We write
* № = ». J f(Y)U(r)dY+y„ f f(Y)U(r)dY,
Q n K ( X , 3 R ) Q \ K ( X , 3 R )
«’,(-*) = У. J f(Y )D xlU(r)dY+ya f f(Y )D xtU(r)dY (ieN).
Q n K ( X , 3 R ) Q \ K ( X , 3 R )
Let X e K ( X , R). For Y e K ( X , 3R) we have r < 4R. For Y e Q \K (X , 3R) we have r > 2R. Hence by (13) we get
(14)
V ( * ) l ^ Mi J r~sd Y + M 2 j \f(Y)\dY when n = 2,
К ( Х , З К ) n
-j |w(X)| j r~" + 2d y + M 2 J |/(Y)|dY when n ^ 3,
K ( X , 3 R ) Q
J г-”+^ Г + М 2Л/(ЛМУ (ieN), '
K ( X , 3 R ) П
for X e K ( X ,R ) ; M 1, M 2 are positive constants.
By (14) and by assumptions of Lemma 4 we get the local uniform convergence of the integrals w(X), wf(X) (ieN) at X e Q . 2° is a consequence of 1°.
Lemma 5. I f the functions f( Y ) , Dyif ( Y ) (ieN) are continuous and bounded in Q and jol/(Y)|dY < oo, then
1° vv(Jf) is of class C2 in Q;
2° vv(AT) satisfies equation (2) in Q.
P ro o f. Suppose 3rj > 0, 0 < rj < R, and K(X, 3rj) c Q. We write
w(X) = L,(X) + L 2(X),
where
В Д = Т. 1 f(Y)U (r)dY,
K(X,3ri)
L2(X) = y„ J f(Y)U(r)dY.
n \ K ( X , 3 t j )
Let XgK( X, ri); for Y e Q \K (X , Ъг\) we have r ^ 2»/. Thus by (11), L2(X)' is of class C2 in X (Z, rj) and its derivatives up to order two may be found by differentiation under the integral sign. Taking into consideration the above properties and the fact that U(r) as a function of X (X Ф 7) satisfies equation (1) we have
(15) (A +c2)L2(X) = 0 for X e K ( X , rj).
Using the formula DxtU(r) = —DyiU(r) (ieN) and Lemma 4, we get Dn L 1( X ) = - y, J f(Y)D „U (r)dY (ieJV).
K( X,3tt)
Integration by parts gives
(16) . DxlL A X ) = yn J Dyif(Y )U (r)dY
K ( X , 3ti)
- I n J f(Y)U(r)cos{nY, yJdSy
dK(X,3tj)
for X eK(X, rj) (ieAT), where nY denotes the exterior normal to ÔK(X, 3rj). It follows from (16) that L t (X) is of class C2 in K(X, rj) and
D i,Ll (X) = yn J Dn f ( Y ) DMU(r)dY
K( X,3t\)
+ Уп j f(Y)U (r)cos(nY, y^dSy
d K (X ,3 q )
for X e K ( X , »/), i eN. Hence
(17) L 1(X) = B 1(X) + B2(X)
where
Bt (X) = y. J t DxlU(r)Dnf(Y )d Y
K(X,3rt) i = 1
B2(X) = y. f f ( Y ) D nvU(r)dSy
8K(X,3ri)
for X e K ( X , rj). Since on the boundary of the ball K(X, 3rj) we have DnrU{r) = Df U(r) |r=3„
we get by (17)
(18) L l (X) + c2L l (X)\x =x — B W + B ^ + c tL J X )
where
B2(X) = y„D,U(r)\t , 3, S f(Y )d S y.
дК(Х,Зф
The integrals By(X) and Ь ±(Х) are locally uniformly convergent at X and hence
(19) B ^ X ) ^ 0, L yfX)->0 as t/->0.
By (12) it is easy to observe that
(20) B2( X ) ^ f ( X ) as f / - 0 . By (15), (18), (19), (20) we obtain
(A + c2)w(X)\XmS = (A+c2)L 1(X) + (A+c2)L2(X) = f ( X ) for X e Q . The proof is complete.
Consider the functions
T,;(X) = г Л / т в д с „ ( х , Y )-U (r)]dY Q
where q> = (<p15 ..., (р^еФ, is given by (6), |a| = 0, 1, 2. Set T,(X) = 7j0--°>(*).
We shall prove the following
Lemma 6. I f the function f is measurable and bounded in Q and Jo I f(Y )\d Y < oo, then
1° the integrals T“ are locally uniformly convergent at every point X e Q ; 2° the function T^X) is of class C2 in Q and
D'xT^X) = T£(X) for X e Q ; 3° the function T^X) satisfies equation (1) in Q.
P ro o f. The function G^ — U is a linear combination of the functions Ua and I(Ua,) where I(Va.) is given by (3), a e B \ { ( 0 , ..., 0)}, a'eA(n1}..., nk).
Hence in order to prove 1° and 2° it is enough to prove the local uniform convergence at every X e Q of the integrals of the form
J°<(X) = $ f(Y )D “x U J Y n
and (by Lemma 1)
K (X ) = f f(Y)ITx l(Ua) d Y = \ f { Y ) H m V a)dY,
n n
where |a| = 0 , 1,2. Suppose K(X, q) cz Q. Then there exists ô > 0 such that ra > à, ra, ^ b for X e K ( X , rj), Y eQ , (vni, ..., u J e R n+1(...,nk. Hence by (11) we have \D«x Ua\ ^ M 3, \D\Ua\ ^ M 3 for X e K (X ,r ,), Y eQ , (vni, v nk)
еКл+1,...,пк, where М 3 is a positive constant. The above inequalities imply the local uniform convergence of J*(X) (i = 1, 2; |a| = 0, 1, 2) at Xe Q. Since by Lemma 3 the function Gv — U satisfies equation (1) as a function of X e Q with Y e Q fixed, X Ф Y, we get
(A + c2)T JX ) = y . [ f ( Y ) ( A + c2)(Ge,- U ) d Y = 0 for Xe Q.
Q
This completes the proof of Lemma 6.
Le m m a 7. I f the function f is bounded and measurable in Q and Jo I f(Y )\d Y < oo, then
1° the integrals J o / (Y)DXiGq>(X, Y)dY when (pi = l , the integrals Jn f ( Y ) G (p( X , Y)dY when <pf = 2, and the integrals JQf{Y)(DXi + h)G(p(X , Y)dY when (Pi = 3 are locally uniformly convergent at every X e S f (i e N );
2° the function иф(Х) given by (8) satisfies the boundary conditions (9).
P ro o f. Since Gy is a linear combination of the functions Ua and I(Ua>), a e B ,a 'e A \B , by Lemmas 4 and 6 and by 3° (c) of Lemma 2, in order to prove 1° it is enough to show that the integrals
(21) jf( Y )D iU .d Y , f f(Y)H (D }U ,)dY,
a о
where a e B \{(0, ..., 0), e j , a' e A(n1, ..., nk) and (nt , . . . , nk) is a subsequence of ( 1 ,..., i — 1, i + 1 , ..., n), |a| = 0, 1, are locally uniformly convergent at every point X e S * ( i e N ) . Suppose X e S f and the projection of a ball K ( X , rj) on St is in S f . Then there exists <5 > 0 such that ra ^ 5, ra>^3 for X e K ( X , rj) n(ÜKj Sf), Ye Q, (vni, . . . , v„k)e R n+,... „k. Hence by (11) and by assumptions of Lemma 7, the integrals (21) are locally uniformly convergent at X. From 1° of Lemma 7 and 3° of Lemma 3 we obtain assertion 2° of Lemma 7.
The following theorem is an immediate consequence of Lemmas 5-7.
Th e o r e m 1. I f the functions f ( Y ) , f y.(Y) (ieN) are defined, continuous and bounded in Q and |/ (Y)\dY < oo, then the function uv(X) given by formula (8) is a solution of (2) in Q satisfying the boundary conditions (9).
References
[1] M. F ila r , Boundary value problems for the Helmholtz equation in a rectangular polyhedral angle o f the Euclidean n-space, Comment. Math. Prace Mat. 24 (1984), 31-41.
[2] N. N. L e b ie d ie w , Special Functions and Their Applications, PWN, Warszawa 1957 (in Polish).
INSTITUTE O F MATHEMATICS, ACADEMY O F MINING AND METALLURGY A. MICKIEWICZA 30, 30-059 KRAKÔW, POLAND
