QUARTERLY OF APPLIED MATHEMATICSVOLUME LI, NUMBER 1MARCH 1993, PAGES 23-35
ON HELMHOLTZ'S DECOMPOSITION THEOREM ANDPOISSON'S EQUATION WITH AN INFINITE DOMAIN
By
TON TRAN-CONG
Aeronautical Research Laboratory, Port Melbourne, Australia
1. Introduction. The solutions of many problems of mathematical physics dependdelicately on the applicability of the classical Helmholtz [1] (also called Stokes [2]-Helmholtz) decomposition theorem. The applicability of this theorem is extendedhere, using a modified form of the solution to Poisson's equation, from that of thecurrently known version, and the theorem is also generalized into an //-dimensionalversion.
The theorem is known as the fundamental theorem in vector analysis (Sommerfeld[3, p. 147]) and states either that every arbitrarily given 3-dimensional vector func-tion u(x) (subject to some condition of differentiability) can be decomposed intoa curl-free vector plus another divergence-free vector (weak version), or that it canbe decomposed into the gradient of a scalar function plus the curl of another vectorfunction, i.e.,
u(x) = V0 + Vxb (1)2(strong version). The theorem can be proved using the identity V w = V(V • w) —
V x (V x w) where w(x) satisfies V w = u. However, such a simple proof, as givenin vector analysis (see, e.g., Lass [4, p. 156], Aris [5, p. 70], or Bowen and Wang [6,
2 sp. 328]), requires u(x) or V • u to be of order <9(|x| ), 8 > 0, at infinity whenthe region DcJi3 under consideration is infinite. Various authors have attemptedto avoid or relax such restriction. Phillips [7, p. 186], and Weatherburn [8, p. 74],used a more complicated application of the solution to Poisson's equation to relax
i c
the restriction to |u(x)| = 0(|x| ), S > 0, at infinity. Blumenthal [9] deviseda method of accelerating the convergence of the solution to Poisson's equation andproved that every function u(x) e C°°(D) bounded at infinity by 0(log|x|) can bedecomposed into a curl-free vector and a divergence-free vector, which are also inC°°(D). Gurtin [10] applied the method to prove that every u(x) e C°(D U dD) nCl(D - dD) bounded at infinity by (9(|x|~<i), <5 > 0, can be written as V8 + V x bfor some 0(x), b(x) e c'(D - dD). We also have another line of approach tothis decomposition problem (Nikodym [11], Friedrichs [12], Weyl [13], Bykhovskiand Smirnov [14], and Fujiwara and Morimoto [15]), complementing the classical
Received May 15, 1990.1991 Mathematics Subject Classification. Primary 31A30, 31B10, 53A45, 58A10.
one just mentioned, that uses functional analysis on those functions of the spaceLr, (1 < r < oo). The r th powers of these functions also have to decay at infinity.Blumenthal's and Gurtin's results remain until now the least restrictive of the directlyderived versions of the theorem. For examples of their use and the significance ofthe above discussed restriction in elasticity and fluid mechanics, see the works byGurtin [10, 16], Millar [17], Hirasaki [18], Aregrebesola and Burley [19], Richardsonand Cornish [20], and Morino [21],
Since the restriction originates from the solution to Poisson's equation with aninfinite domain, it is expedient to deal with this equation, which is
V2(f)(\) = i//(\), (2)
where y/(\) e C°(D), D c RN , is an arbitrarily given function. Its classical solutionis (Kellogg [22] for N = 3, Courant and Hilbert [23] for jV > 3)
m ° (aT- W* L iy (y) dr(y)■ w-3- (3)and is applicable to an infinite region D c RN only when y/{x) = 0(|x|~2_(5),S > 0, at infinity. This seemingly minor restriction is the cause for the restriction inHelmholtz's theorem and is also an obstacle to the solution of many other problems inmathematical physics (e.g., the second example of Sec. 5). This makes the relaxationof the restriction a worthwhile effort with ramifications in mathematical physics.
As Blumenthal changed the weighting factor in the integral of Eq. (3) from 1 /|y—x|to (l/|y-x| — l/|y|) so that y/(x) needs only be of order 0(|x|~1~'5) at infinity, wecan extend his process to higher-order terms so that ^(x) now needs to be boundedat infinity only by 6>(|x|"), a a constant. (Obviously, we can subsequently assumea > 0 to be an integer /.) D can be simply or multiply connected. The extensionis similar to the technique used in analysis to prove Carleman's inequalities (seeSchechter and Simon [24], Amrein, Berthier, and Georcescu [25], and Jerison andKenig [26]). The result here is not as general as that given in Hormander's book [27,Corollaries 10.7.10 and 10.8.2], which does not impose any limit on the growth rateof i//(x) and only requires D to be F-convex for singular support, but, owing to itsdirect derivation, should be appealing.
The solution to Poisson's equation so directly derived gives a less restricted, strongversion of Helmholtz's theorem so that u(x) now needs to be bounded at infinityonly by 0(|x|'), I a constant (Theorem 2). D can be multiply connected andmay have internal surfaces. The generalisation of Helmholtz's theorem into an N-dimensional version is realised using the calculus of differential forms. It appears thatit has not been previously given as an analogue of the 3-dimensional version. (The
2 $previous tensorial work by Fosdick [28] requires that |u(x)| = 0(|x| ), S > 0,at infinity and does not correspond directly to the 3-dimensional results.) Althoughthe solution to Poisson's equation can easily give the weak version of the theorem,the strong version cannot be simply derived from the weak one via the Converse ofPoincare's Lemma unless D is topologically very simple, such as being star-shaped,which is a very restrictive situation.
HELMHOLTZ'S DECOMPOSITION THEOREM AND POISSON'S EQUATION 25
Three examples are then given in Sec. 5. The first enlarges the validity of thePapkovich-Neuber general solution to Lame's equation in linear elasticity by relaxing
c
the requirement that |u(x)| = 0(|x| ), 5 > 0, at infinity (e.g., Gurtin [10, 16] andMillar [17]). The second solves the Legendre equation. The third shows that everyharmonic function can be expressed as the divergence of another harmonic vectorfunction; the expression has been found useful in continuum mechanics (e.g., seeTran-Cong [29] and Tran-Cong and Blake [30]).
The layout of this paper is to first establish the solution to the TV-dimensionalPoisson's equation with an infinite domain. Helmholtz's theorem is then given withits 3-dimensional corollary. The application finally follows.
2. Definitions and notations. This paper deals with an TV-dimensional space, TV >3 . A letter will denote a scalar quantity when it is non-bold and an TV-dimensionalvector or tensor when it is latin and boldface. The letters i, j, k, I, m , n denotenonnegative integers. The symbol K denotes the set A" = {1,2,...,TV-1,TV},[/] = [1, 2, , i, ... , TV] e Kn~1 an increasing (TV - 1 )-tuple with i absent fromthe tuple, and [ij] = [1, 2,, i,, j, , TV] e KN~2 an increasing (TV - 2)-tuple with both i and j absent (i < j always). The symbol R = (-oc, oc) denotesthe set of all real numbers, S{a, p) = {x | |x - a| < p) an open sphere centered on ahaving radius /?>0,and S(sl, p) = {x| |x-a| < p] its corresponding closed sphere.
The permutation symbol 8n "N has the value 1, -1, or 0 depending on\ 2 " N
whether (/, , i2, ... , iN) is an even, odd, or not a permutation of (1 , 2, ... , TV).The Kronecker symbol Sj has the value of unity when i — j and zero other-wise. Vx and dx denote + e2d/dx2 + ■ ■ ■ + eNd/dxN and e, dxx +e2 dx2 + b eN dxN , respectively, where (e,, e,, ... , e^) are the basis of the co-ordinate system {xx, x2,... , xN). When no confusion can arise, the subscript ofV may be omitted. With a £-tuple i = (/,, /2, ... , ik) attached to x, we definedx. = dx e A dx, e, A ■ • ■ A dx, e, = dx; dx, ■ • dx e A e A • • • A e , where A
1 1 'l '2 l2 k k 1 '2 k l\ l2 lk
denotes the exterior product (Bowen and Wang [6, p. 303]).Let /(u) be a scalar function of a vectorial variable u = uN). Its
differential dkf(a, b), where k > 0, is a function defined by
//(a,b)= ^ (- f/(u). ) blbl -bi, k> 0.z—' \ du■ du • • • du / 'i '2 h
1</,, i2,..., ik<N \ '1 '2 h / u=a
When c(x) is a tensor, dc denotes its exterior derivative (Bowen and Wang [6, p.303]); this should not be confused with the notation for the differentials.
The constant k and functions h(x) and g(i, y, -x), i > 0, denote
K = T{N/2)[(N -2)27tm]~l (4)
andh(x) = k/\x\N"2, TV > 3, (5)
andi
a ( i v —= h I\j — — \k\g(i, y, -x) = h(y-x)-J2^dkh{y, -x), (6)
k=0
26 TON TRAN-CONG
respectively, where T(x) is the gamma function with argument x .Definition 1. A domain is an open set, any two of whose points can be joined
by a polygonal line, of a finite number of segments, all of whose points belong to theset (Kellogg [22, p. 93]). A region D is either a domain, or a domain together withsome or all of its boundary points (Kellogg [22, p. 93]). The term regular region isdefined as in Kellogg's book [22],
Definition 2. A function /(x) defined in D satisfies a Holder condition withexponent a (a > 0) at a point a if there are two positive constants c(a) and M{a)such that |/(x) - /(a)| < M(a)|x - a|" for any x such that |x — a| < c(a).
dD and D — dD denote the boundary and the interior of D , respectively, /(x) eCn(D) denotes that the function /(x) is defined, continuous together with all of itspartial derivatives of order up to and including n (« > 0) in the region D , /(x) €C"'a(D) denotes that /(x) e Cn(D) and all of its n th order partial derivativessatisfy the Holder condition with exponent a in D .
Every k th-order (0 < k < N) antisymmetric tensor b(x), which changes signwhen any two of its indices are interchanged, corresponds to a differential &-formb(x) defined by (Bowen and Wang [6, p. 303])
b(x) = V b , dx e A dx e A •••A dx ev ^ hh -'k 'i !i <i h 'k lk
bi i i dxZ—^ /, h'"h1 < I .</,<■■
k 'l'2
The symbol (b), , , , 1 </,</-,<•••< i, < N, denotes the component b, , ,l\l2"/k 1 z K l\l2"'lk
of the antisymmetric tensor and the differential /:-form b.Definition 3. A differential £-form b (1 < k < N) is exact if it is equal to the
differential of another differential (k — l)-form c, i.e., b = do..
3. Solution to Poisson equation in an infinite domain. We first note that there is aconstant c > 0 such that the function u(x) defined by
u{x) = <
1 for x > 1,2
0 for x < - j, ^
-l 2exp(l/(4? -1 ))dt otherwise1/2
is in C°°(R). If y/(x) e C"'"(D), 0 < a < 1 , then we have [u(x)y/(x)] e C"'a{D).We next consider the Laplacian of the {k + 2)th differential of /(y):
v;(/+2/(y,x))= EijeK (dxf
d(dk+\f(y, x))' dy.
2 y- sj d2(dk+lf(y, x)) d\dk+lf{y, x))^ ' dxByi dyjdxidxii.jeK i ■'J i.jek -J ' i
HELMHOLTZ'S DECOMPOSITION THEOREM AND POISSON'S EQUATION 27
But
y —
y —1 </,«,. «2 ik+l<Ndyi
= (k+l)V2Jdkf(y, x)).
dk+lf{y) a ,dyj dyt ■■■dyi flx^'A "
l\ h lk±\ 1lk+1
f/:in[ d m Ixx. •••X. d'k+iI dy dy ■ ■ ■ dy I '> '2 '* '\ l\ h ^ if 4-1 /
in D U dD . It is also twice differentiable and satisfies in D — dD
d2 ,, , Sj , s f , , d2<t>{x) = ~H/{x) -dx-dx, N lim f V(y)-),f {^(y - X)[J - "(|y| - 2)]ct—o Jd-s(x,o) axjox] j+ g{l + 2, y, -x)w(|y| -2)}dr(y) 1 < i,j < N,
(17)and
V>=y/. (18)For every S(q, rj) C {D - dD) and any integer n > 0 it satisfies
</»(x)eC"+2(S(q,v)) if f/(x)eC"+'(5(q, r,)). (19)Proof. Let D, = Dn5(0, 3), D-, = D - S(0, 1), and <j/(\) = (//,(x) + i//2{x)
where y/,(x) = [1 - m(|x| - 2)]^(x), y/2(x) = u(\x\ - 2)]y/(x), and <//,(x), i//2(x) eC°{DudD)nC°-a(D - dD).
We note that dkh(y, -x) e Coc(Z)" x RA) for all k > 0, that ^,(x) = 0 outside£>, n 5(0, 5/2), and that <j/2(x) = 0 outside D2 - 5(0, 3/2). We then have 0(x) =0,(x) + <t>2(x) where
0, (*) = — / ^(y-*)V,(y)^T(y) and <f>2(x) = - g(l + 2 , y, -x)y/,(y) dz{y).Jd, Jd,
HELMHOLTZ'S DECOMPOSITION THEOREM AND POISSON'S EQUATION 29
Since Z), is finite, only 02(x) needs to be considered. Taylor's theorem givesg(l + 2, y, -x) = (dl+ih(y - tx, -x))/(/ + 3)! where 0 < t < 1 . As |y| —» oo,g(l + 2, y, -x) = Odyl ^-'-1) and y/(y) — 0(|y|'); therefore, the integrand is oforder 0(|y|—jV—1) at infinity and <^>2(x) is defined for all x e {D - dD).
We have to establish the rules for the first and second derivative of </s(x) for allx € D and also the equality
-vj| [ g{l + 2, y, -x)y/2(y)dr{y) = y/2{x)Jd,
(20)
since g(l + 2, y, -x) is singular and D2 is unbounded. We can either follow themethod in Kellogg's book [22, pp. 150-156] or apply Harnack's convergence theorem.The latter approach is shorter and is adopted here.
Choose an arbitrary point q 6 (D - dD). As D2 overlaps D n S(0, 3/2), thereis an rj > 0 such that at least 5"(q, rj) c D, or S(q, rj) c (£Tl5'(0, 3/2)). We thensuppose that ^(x) e C"+1(S(q, t])).
We first suppose that S(q, tj) c D2, hence y/2{x) e C"+1(>S(q, r})). Take anarbitrary point w e 5(q, r]) and choose a constant X > 0 such that S(w, X) cS(q,rj). The differentiability of </>2(x) in S(w,X) will be examined. Write 4>2(x)as the sum of two integrals, </>21(x) and </>22(x), of ^2(x) over an<^ ~~5(w, A), respectively. The first,
r 1+2021(x) = - / [h(y - x) /h{y, -x)]y/2{y)dr(y),
Js(w, A)
is over a bounded region with a singular kernel caused by |y - x\2~N and is cov-ered by Lemma 2 and the results of Lemma 1 regarding ~Xdkh(y, -x).Hence, </>21(x) satisfies Eqs. (16) to (19) with D replaced by S(w, A) and{h(y — x)[l - w(|y| - 2)] + g(l + 2, y, -x)w(|y| - 2)} by g(l + 2, y, -x)w(|y| - 2).In particular, we have
Vx<Mx) = ~vx f g(l + 2> y» -x)V2(y)dr(y)JS( w,A)
= -V;/ /j(y-x)^2(y)c/r(y)J S(w, A)
-yx/ -x)V2(y)vfc=0 /c! /
= y/2(x) - f vlfjz hdkfl(y> ~x))il/2(y)dz(y) = y/2(x)- (21)
Differentiation has been brought under the fourth integral sign since the summation1+2 ,
i//2(y) d h(y, -x) is infinitely differentiate with respect to x and thesek=0
derivatives are continuous for (x, y) e S(w, A) x S(w, A) where the set is compact(Apostol [36, p. 167]). We also note that </>21(x) e C°{D\JdD)nCn+2{S(w, A)). The
30 TON TRAN-CONG
second integral 4>22(x) is over an unbounded region but with a bounded, infinitelydifferentiable kernel and is considered below.
Let {pm} be an unbounded increasing sequence with /?, > |q| + t]. The function
= g(l + 2, y, -\)i//2{y)dz(y) (22)JD2nS(0,pJ-S(v,l)
is harmonic for x e S(w, A/2) by Lemma 1. £m(x) can be differentiated under theintegral sign since g(l + 2, y, -x) is infinitely differentiable with respect to x andthese derivatives are continuous for (x, y) e S(w, A/2) x ((D1 U dD,) n 5(0, pm) -.S(w, A)), where the set is compact.
Let x e S{w,A/2) and let m —> oo; then pm also tends to infinity. Since\g(l + 2, y, -x)^2(y)| is bounded for large |y| by A^lyl-A —1 with some M inde-pendent of (x, y), the sequence of harmonic functions {£m(x)} converges uniformlyon S(w, A/2) to the limit function </>22(x) £ C°°(S(y/, A/2)) given by
22 » = -/ g{l + 2, y, -x)^2(y)</r(y). (23)•/£>,—S(w, A)
By Harnack's convergence theorem (Kellogg [22, p. 248]), £m(x) together with allof its derivatives converge on S(w, A/2) to </>22(x) and its respective derivatives and</>22(x) is harmonic in S(w,A/2). Hence
[ vx<?(/ + 2, y. -x)^2(y)<a'r(y)^D2nS(0,pm)-5(w,A)
Vx^(/ + 2, y, -x)^2(y)rfr(y), (24)/D2-S(w,A)
:^(x) 3 - 5Z53T " " JSS, f 5^-"")2
*°° JD2ns(0, pm)-S(Y, A) dxfixj
= [ f) x rt y S(l + 2,y, -X)^2(y)rfi(y), (25)JD2-S(*r.X) OXfiXj
and
-V>22(x) = lim f \72xg(l + 2, y, -x)^2(y) flfr(y)m^ooy£>2n5(0,/>,„)-5(w,A)
= lim 0 = 0, (26)m—>oc
by Eq. (9) and Lemmas 1 and 2. Differentiation has been brought under the integralsign for the third members of Eqs. (24) and (25) since g(l + 2), y, -x)^,(y) isinfinitely differentiable with respect to x and these derivatives are continuous for(x, y) e S(w, A/2) x ((D2 U 8D2) n 5"(0, pm) - S(w, A)) where the set is compact.Since w is an arbitrary point of S(q, rj), it follows that 022(x) is harmonic in
HELMHOLTZ'S DECOMPOSITION THEOREM AND POISSON'S EQUATION 31
S(q, t]). Consequently, 022(x) e C°°(.S(q, t])) and the differentiability of </>21(x)determines that of </>2(x). Finally, since <^>22(x) is harmonic, Eq. (21) gives
Hence 0(x) = </>,(x) + </>21(x) + </>22(x) satisfies Eqs. (16), (17), (18), and (19).Secondly, we can have a similar but simpler argument when S(q, ?/) c (Dfl
5(0, 3/2)) since 021(x) = 0 in this case. The theorem has been proved. (It couldalso be generalized to state that if i//(x) € C°(DudD) then 0(x) e C\Dl\dD) withonly some simple change of the proof.)
Lemma 3. Let a(x) be a twice differentiate vector function of x. Then the(N - 2)th order antisymmetric tensor b(x) defined by
satisfies= (29)
This is the A^-dimensional generalization of V(V • a) — V2a = Vx(Vxa).Proof. The (N - 2)-form
b(x) — J2 (30)iimKN-2
has for its exterior derivative db the (N — l)-form
,, , , si'-l ^^\2--N J . t .sj-2®b\2■■■i-.-j-.-N 1^b= E (-1) »7^dxm+ E (-1) —~^T~ Ordxt ' ^ dxjBier-' Uj]eK 1
Define
Bij - ~Bji - bn-i-j-N for ' < J> Bij = 0 for ' = J' (31)
and note that Btj = (-1 )'+J(daj/dxj - dajdx■) for all 1 < i, j < N. We have
wh V*V\ n-'^w n'-V d (da> da\ a<n>=£D-') (-') rxi'Ti=l j=l ' ;=1 j=l i \ • J/
(32)which gives Eq. (29).
Theorem 2. Let D c RN be an infinite regular region and let f(x) e C°{D U dD) nC°'a(D - dD) be a vector function bounded at infinity by 0(|x|'), / a constant.Then f(x) can be written as
Nt jy-i,f(x) = V0(x) + Ee,(-1)7 Vb>l2..j..v (33)
j= i
32 TON TRAN-CONG
in D — dD for some functions 6(x), b(x) e C°(D U dD). For every S(q, rj) C(D - dD) and any integer n > 0, the scalar and the (N - 2)th-order tensorialfunctions 0(x) and b(x) satisfy
6(x),b(x)eCn+\S(q,tj)) iff(x) e C"+I(5(q, f/)). (34)1 2Proof. Theorem 1 gives a solution a(x) e C (D u dD) to V a = f. For every
S(q, r\) c (D - dD) and any /i ^ 0, satisfies
a(x)eC"+2(S(qlf|)) if f(x) € C"+1(5'(q, t])). (35)
Equation (28) defines an (N - 2) th-order tensor b(x) e C°(Du3£>)nC"+1S(q, rj).Lemma 3 then gives Eq. (29). Define 8(x) = V x-a(x) E C°{D U dD) Cn+l S(q, tj)and recall that V"a = f; we have the required Eq. (33). For every S(q, >/) C (D-dD)and any n > 0, we then use Eq. (35) to obtain
0(x), b(x) G C"+'(S(q, »,)) if f(x) e C"+'(5(q, r,)).V6(x) is curl-free. The divergence of the vector
N*(*) =
7=1
is calculated fromN
(V • s) dx{1 N = J2id[(db)l2...j...N]/(dXj)}(-l)J~l dxf j/\ dx{]] = d{db).j=i
The last member is identically zero by direct calculation (Poincare's Lemma; see,e.g., Flanders [37]). The theorem has been proved.
3 0Corollary 1. Let D c R be an infinite regular region and let f(x) G C (DudD)CiC°'a(D - dD) be a vector function bounded at infinity by 0(\x\'), I a constant.Then f(x) can be written as
3f(x) = V0(x) + (db)l2_j_3 = V0(x) + V x b(x) (36)
j=iin D - 3D for some functions 0(x), b(x) € C°(D U dD). For every S(q, rj) c(D-dD) and any integer n> 0, the scalar and the vectorial functions 0(x) andb(x) satisfy
0(x), b(x) e C"+1(5(q, t])) if f(x) e C"+1(S(q, »/)). (37)5. Applications.1. The homogeneous Lame equation is
(1 - 2z/)V2u(x) + V(V -u) = 0 (38)where v is a constant (Poisson's ratio). Mindlin [38] applied Helmholtz's decompo-sition (twice) to u(x) to show that the general (Papkovich-Neuber) solution to Eq.(38) is
u(x) = 4( 1 - u)b - V(x • b + cj>) where V2b(x) = 0 and V20(x) = 0. (39)
HELMHOLTZ'S DECOMPOSITION THEOREM AND POISSON'S EQUATION 33
Theorem 2 shows that the solution is applicable whenever |u(x)| = 0(\x\'), I aconstant, at infinity.
2. The solvability of the "Legendre equation", found in the studies of Legendrefunctions,
1 9 j + cosy-^ (^osy~ j\ x +W, y) = b(6, y), (40)cos2 y \dd
where b(6, y) is a given function, is a central device in the proof of the general-ity of the above Papkovich-Neuber solution in terms of only b(x) (i.e., withomitted; see Tran-Cong [39]). Its solution is complicated by any other method butis fairly simple using the two-dimensional analogue of Theorem 1: By setting rj —In |[ 1 + tan(y/2)]/[ 1 - tan(y/2)]|, a{6, rj) — b(6, y), it becomes
( d2 d2 \ 4e2n 4e2'1(s?rte-<41)
Define x = {xx,x2) = (d, tj), h2{x) = (4n)~l ln(x2 + x2), g2(l + 2,y,-x) =/j2(x-y) - _x)- The anal°gue of Eq. (15),
where K{\, y) = [87re2xV(e2*2 + l)2]{/*2(x-y)[l-u(|y|-2)]+g2(/+2, y, -x)«(|y|-2)}and F(x{, x2) = [4e2*2/(e2Xl + l)2]a(Xj, x2). The solution £(x,, x2) to Eq. (43)is periodic in x{ = 9, and Eq. (42) specifies the correspondence between £(x) and£(x) ■
3. Consider next another interesting application of Theorem 1 and the techniqueof Theorem 2.
Theorem 3. Let the infinite regular region D c RN be star-shaped, which has theproperty that every point in D can be joined to the origin by a straight line segmentthat lies totally in D, and let i//(x) e C°{D u dD) n C°°(£) - dD) be a harmonicfunction in D Li dD and bounded at infinity by 0(|x|7), I a constant. Then t//(x)can be expressed as
^ = V- a where V2a = 0 and a(x) e C°(D U dD) n C°°(D - dD). (44)
Proof. Consider the (N - l)-form c = {dyz/dx^dx^. Since
s(x) = - f {h{y-x)[l - "(|y| - 2)] + g{l + 2, y, —x)w(|y| - 2)}b(y) dr(y).Jd
Then the function a(x) = V(f> — e.(—l)7_1(rfs)12...j-...N satisfies
V-a = V^-(rf(rfs))12...JV = V2^ti/.
We also haveV2a = V(V20) - eZ-ir'trffv's))!,.,.,, - 0,
since V2</> = y/ and V2s = b. Therefore a(x) 6 C°(D U <9Z)) n C°°(D - <9D).The theorem has been proved. It can even be generalized so that D needs only bedeformable to star shape.
When N = 3, the theorem states that every harmonic function ^(x) in a regionD c R3 with m internal surfaces Sx , S2 , ... , Sm is expressible as
a\f/(\) = J^2i- —r + V-a where V"a = 0, (46)
k=i |X~C*'
with ck and qk , 1 < k < m, being, respectively, the constant position vector andthe associated scalar constant of an interior point of the corresponding internalsurface Sk . D need not be star-shaped for N = 3 since Stevenson's proof [40],instead of the Converse of Poincare's Lemma, is used to write V = V x b +J2k=l Qk(ck ~~ x)/lc/t ~ x|2 ■ The gradient of Eq. (46) also gives the first part of Theo-rem IV of Weyl's paper: The gradient of every harmonic function i//(x) is equal toV X b + Y!k=\ Qk(ck ~ X)/IC£ - x|2 for some b(x).
References
[1] H. Helmholtz, Ueber Integral der hydrodynamischen Gleichungen welche den Wirbelbewegungenentsprechen, Crelle 55, 25-55 (1858)
[2] G. G. Stokes, On the dynamical theory of diffraction, Cambridge Philos. Trans. 9, 1-62 (1849)[3] A. Sommerfeld, Mechanics of Deformable Bodies, Academic Press, New York, 1950[4] H. Lass, Vector and Tensor Analysis, McGraw-Hill, Kogakusha, Tokyo, 1950[5] R. Aris, Vectors, Tensors and the Basic Equations of Fluid Mechanics, Prentice-Hall. Englewood
Cliffs, NJ, 1962[6] R. M. Bowen and C. C. Wang, Introduction to vectors and tensors, Vectors and Tensor Analysis,
Vol. 2, Plenum Press, New York, 1976[7] H. B. Phillips, Vector Analysis, John Wiley & Sons, New York, 1933[8] C. E. Weatherburn, Advanced Vector Analysis, G. Bell & Sons, London, 1924[9] O. Blumenthal, Uber die Zerlegung unendlicher Vectorfelder, Math. Ann. 61, 235-250 (1905)
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[31
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[34
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[36[37
[38
[39
[40
HELMHOLTZ'S DECOMPOSITION THEOREM AND POISSON'S EQUATION 35
M. E. Gurtin, On Helmholtz's Theorem and the completeness of the Papkovich-Neuber stress func-tions for infinite domains, Arch. Rational Mech. Anal. 9, 225-233 (1962)O. Nikodym, Sur un theoreme de M. S. Zaremba concernant les fonctions harmoniques, J. de Math.12, 95-108 (1933)K. Friedrichs, On differential operators in Hilbert spaces, Amer. J. Math. 61, 523-544 (1939)H. Weyl, The method of orthogonal projection in potential theory, Duke Math. J. 7, 411 -444 (1940)E. B. Bykhovsky and N. V. Smirnov, On orthogonal expression of the space of vector functionswhich are square-summable over a given domain, Trudy Mat. Inst. Steklova 59, 5-36 (1960)D(aisuke) Fujiwara and H(iroko) Morimoto, An Lr-theorem of the Helmholtz decomposition ofvector fields, J. Fac. Sci. Univ. Tokyo Sect. I Math. 24, 685-700 (1977)M. E. Gurtin, The linear theory of elasticity, Mechanics of Solids, vol. 2, edited by C. Truesdell,Springer-Verlag, Berlin, 1973R. F. Millar, On the completeness of the Papkovich potentials, Quart. Appl. Math. 41, 385-393(1984)G. J. Hirasaki, Boundary conditions on the vector and scalar potentials in viscous three-dimensionalhydrodynamics, Quart. Appl. Math. 28, 293-296 (1970)Y. A. S. Aregrebesola and D. M. Burley, The vector and scalar potential method for the numericalsolution of two- and three-dimensional Navier-Stokes equations, J. Comp. Phys. 24, 398-415 (1977)S. M. Richardson and A. R. H. Cornish, Solution of three-dimensional incompressible flow prob-lems, J. Fluid Mech. 82, 309-319 (1977)L. Morino, Helmholtz decomposition revisited'. Vorticity generation and trailing edge condition,Comp. Mech. 1, 65-90 (1986)O. D. Kellogg, Foundations of Potential Theory, Springer, New York, 1929; Dover, New York,1953R. Courant and D. Hilbert, Methods of Mathematical Physics, Interscience, New York, 1962M. Schechter and B. Simon, Unique continuation for Schrddinger operators with unbounded poten-tials, J. Math. Anal. Appl. 77, 482-492 (1980)W. O. Amrein, A. M. Berthier, and V. Georcescu, Lp-inequalities for the Laplacian and uniquecontinuation, Ann. Inst. Fourier (Grenoble) 31, 153-168 (1981)D. Jerison and C. E. Kenig, Unique continuation and absence of positive eigenvalues for Schrddingeroperators, Ann. of Math. (2) 121, 463-494 (1985)L. Hormander, The Analysis of Linear Partial Differential Operators II, Springer-Verlag, Berlin,1983R. L. Fosdick, On the vector potential and the representation of a polyharmonic function in n-dimensions, J. Math. Mech. 14, 573-587 (1965)T(on) Tran-Ccng, Notes on some relationship between the Galerkin, Papkovich-Neuber and Naghdi-Hsu solutions in linear elasticity, Mech. Res. Comm. 8, 207-211 (1981)T(on) Tran-Cong and J. R. Blake, General solutions of the Stokes's flow equations, J. Math. Anal.Appl. 90, 72-84 (1982)E. Schmidt, Bemerkung zur Potentialtheorie. Mathematische Abhandlugen Hermann AmandusSchwarz zu seinetn funfzigjahrigen Doktorjubilaum, Springer, Berlin, 1914D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed.Springer-Verlag, Berlin, 1983S. G. Mihlin, Singular integral equations. Translation No. 24, Amer. Math. Soc., Providence, RI,1950A. Calderon and A. Zygmund, On the existence of certain singular integrals. Acta Math. 88, 85-139(1952)M. H. Taibleson, The preservation of Lipschitz spaces under singular integral operators, StudiaMath. 24, 107-111 (1963)T. Apostol, Mathematical Analysis, 2nd. ed., Addison-Wesley, Reading, MA, 1974H. Flanders, Differential Forms with Applications to the Physical Sciences, Academic Press, NewYork, 1963R. D. Mindlin, Note on the Galerkin and Papkovich stress functions, Bull. Amer. Math. Soc. 42,373-376 (1936)T(on) Tran-Cong, On the completeness of the Papkovich-Neuber solution. Quart. Appl. Math. 47,645-659 (1989)A. F. Stevenson, Note on the existence and determination of a vector potential. Quart. Appl. Math.12, 194-198 (1954)
