See whether any terms must be changed in expression (5) in
the notes on elliptic equations
in the three dimensional case. Then determine how (6)
differs from the two-dimensional case.
Assume that you choose
where is a constant still to be chosen and function
are the same function using different names. Show that then on the
surface of the unit sphere,
Show next that
where subscripts indicate partial derivatives with respect to that
variable. Conclude that on the surface on the unit sphere
Now look up the Laplacian in spherical coordinates and evaluate
in terms of
using the chain rule of
differentiation. Show that
Hint: in the right hand side, differentiate out the first product
before comparing with what you got from differentiating out
. Conclude that the Laplacian of
is zero.
In view of the formulae derived in the previous question, show
that must be in order for the surface integrals in (6)
only to involve the function
given in a Dirichlet problem, and not the then unknown function
. Also show that it would not be possible to
choose so that drops out from both integrals, so that the
Neumann problem cannot be solved this way. For the Dirichlet
problem, use spherical coordinates for the point at which
is to be evaluated and for the generic integration point
as in
In those terms, show that the integrals to be evaluated reduce to
Show that the parenthetical expression may be simplified to
Use this plus the made assumption that on the boundary to
derive the Poisson integral (9).
3.38. This does not require solution of the problem using
the Poisson integral formula. You can just examine what symmetry
properties the solution should have to figure out the value
at the origin. However, feel free to check your result against the
Poisson integral.
3.39. Again this does not require solution of the problem
using the Poisson integral formula. You should be able to find the
complete solution by mere inspection. However, feel free
to check your result against the Poisson integral; in that case,
first write the boundary values in the form
The integrals for and can be found in a table of definite
integrals.
3.40. This is about the solution to the Dirichlet problem in a
circle, which, as derived in class, is given by the Poisson integral
formula (also listed in 3.37.) Because the problem for is
linear, the change in due to the change in boundary
condition is the Poisson integral solution when is zero
everywhere except in the interval
, where it
equals the change in boundary condition.