The Fundamental Solution to Laplace’s Equation
Consider the
fundamental solution
to the Laplace operator:
in 3D
and
in 2D
where the variables
x
and
y
represent coordinate positions in space
x
= (
x
1
,
x
2
,
x
3
)
and
y
= (
y
1
,
y
2
,
y
3
)
.
The fundamental solution has the property
in
Ω
where the derivatives are taken with respect to
x
and
δ
is the Dirac delta distribution.
The Dirac delta has the property that when used in an integral it samples the value of a function
f
at
y
:
in 3D
in 2D
in Ω