LECTURE 20
Green's Functions and Solutions of Laplace's Equation, I
In our discussion of Laplace's equation in three dimensions
2 2 2
@ @ @
2
20.1 + + 0= r =
2 2 2
@x @y @z
I p ointed out one solution of sp ecial imp ortance, the so-called fundamental solution
1 1
p
x; y ; z = 20.2 = :
2 2 2
r
x + y + z
Note that due to the singularity at the p oint 0,0,0 , the solution 20.2 is really only a solution for the
3
region R 0; 0; 0 . The nature of this solution when r ! 0isworth examining a little closer.
In terms of spherical co ordinates
p
2 2 2
x + y + z r =
y
1
= tan
x
20.3 p
2 2
x +y
1
= tan
z
wehave
1
r;; =
r
@ @ @ 1 1
^ ^
r = r^ + +
20.4
@r r @ r sin @
2
@ @ 1 @ @ 1 @ 1
2 2
r + sin + r =
2 2
r @r @r r sin @ @ r sin @
^ ^
wherer ^, , are resp ectively, the unit vectors indicating the directions of tangentvectors to the corre-
sp onding co ordinate curves.
2
Applying the gradient r and the Laplacian r to our solution 20.2 we get
@ 1 ^r
^
r = r =
2
@r r r
20.5
@ @ 1 1
2 2
r =0 : r =
2
r @r @r r
However, we should note again that these formula are not really valid when r = 0 since is not continuous
when r =0,we certainly cannot evaluate derivatives of when r = 0 . To study the situation near r =0,
let > 0 b e a small p ositive parameter and de ne
1
= 20.6 :
r +
3
Since r is never negative, is p erfectly regular thoughout R , and obviously
20.7 = lim :
!0 76
20. GREEN'S FUNCTIONS AND SOLUTIONS OF LAPLACE'S EQUATION, I 77
Applying the Laplacian to yields
1 @ 1 @
2 2
r = r
2
r @r @r r +
2
r @ 1
=
2 2
r @r r +
2 2
2r r + +2r r +
1
20.8
=
2 4
r r +
2 2
2r 2r+2r 1
=
2 3
r r +
2
= :
3
r r +
2 3
Now let us now consider the volume integral of r over R .Wehave
R R R R
R 2
2
2 2
r sin dr dd r dV = lim
3 3 R!1
r r + R 0 0 0
R
R
8r
dr = lim
3 R!1
r + 0
R
R+
8
= lim d
20.9
R!1 3
R+
2
4 8
= lim
R!1 2
= 4 :
Notice the result we obtain is indep endent of .
Thus, wehave discovered a sequence of functions f
1
2
f r = r r = 20.10
3
4 2r r +
for which
0 ; ifr 6= 0; 0; 0
20.11 lim f r =
1 ; if r = 0; 0; 0
!0
and for which
Z
20.12 f r dV =1 8 6=0
3
R
But prop erties 20.11 and 20.12 are exactly the prop erties that we demand for a sequence of functions
to de ne a three-dimensional delta-function. See Lecture 7.
3
Indeed, let g r b e a di erentiable function on R and consider the limit
Z
lim g r f r dV :
!0
3
R
According to 20.11 the supp ort of f r for small is concentrated around the origin. For example, if we
set
6
10
6
=10 f r < ; 8 r>1
2
30
and if we set =10
6
10
6
; 8 r>10 : f r <
2
In the limit the supp ort of f r the integrand is precisely the origin O.Thus,
Z Z
lim g r f r dV = lim g O f r dr = g O :
!0 !0
3 3
R R
And so we set
1 1
2 3
20.13 r r = lim
!0
4 r +
20. GREEN'S FUNCTIONS AND SOLUTIONS OF LAPLACE'S EQUATION, I 78
with the understanding that the limit is to b e taken only after integrating. By an abuse of notation one
sometimes writes
1
2 3
20.14 r = 4 r
r
or even more generally,
1
2 3
20.15 r = 4 r r :
o
jr r j
o
Okay, so what is the p oint of all this? Consider the non-homogeneous equation
2
20.16 r = 4g r
1
with g r decaying faster than as r !1. Multiplying b oth sides of 20.16 by
2+
r
1
jr r j
o
3
and integrating over R we get
Z Z
1 4
2
g r dV = r r dV
jr r j jr r j
3 3
o o
R R
Z
1
r = r r dV
jr r j
3
o
R
Z
1
2
r dV = r
jr r j
3
o
R
Z
3
= 4 r r r dV
o
3
R
= 4 r :
o
In the second and third steps wehave used an integration by parts formula coming from Gauss's theorem .
Wethus have the following solution to 20.16
Z
0
g r
r = dV :
0
jr rj
3
R
1
0
Note how the integral kernel G r; r = is used to construct the solution r directly from the \source
0
jr rj
function" g r . More generally,anintegral kernel that interp olates b etween source functions inhomogeneous terms and solutions of a nonhomogeneous PDE is referred to as Green's function for the PDE.