Ross Ogilvie 21st October,
2024
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23.
- Subharmonic Functions
Let
be an open and connected region. A continuous function
is called subharmonic if for all
and
with
it lies below its spherical mean: .
-
(a)
- Let
be two subharmonic functions. Show that
is subharmonic.
(1 point)
-
(b)
- Suppose that
is twice continuous differentiable. Show that
is subharmonic if and only if
in .
(3 points)
-
(c)
- Prove that every subharmonic function obeys the maximum principle: If the maximum of
can be found inside
then
is constant. (2 points)
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24.
- Liouville’s Theorem
Liouville’s theorem (3.10 in the script) says that if
is bounded and
harmonic, then
is constant. In this question we give a geometric proof in
using ball means
defined when
through
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Theorem 3.5 states that if
is harmonic then it obeys .
This beautiful proof comes from the article Nelson, 1961.
-
(a)
- Consider two points in the plane
which are distance apart. Now
consider two balls, both with radius ,
centred on the two points respectively. Show that the area of the intersection is (2 bonus points)
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-
(b)
- Suppose that is a bounded
function on the plane:
for all and
some constant .
Show that (2 points)
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-
(c)
- Let
be a bounded harmonic function. Complete the proof that
is constant.
(2 points)
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25.
- Weak Tea
As in the script, for any test function ,
define a test function
by
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-
(a)
- Describe the support of
in terms of the support of .
(1 point)
-
(b)
- We have defined the spherical mean of a distribution using the formula
for .
Compute
for
and .
Does it have weak mean value property? (3 points)
-
(c)
- Take a function
and fix a radius .
Consider the function .
Prove
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(2 points)
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(d)
- The spherical mean for distributions, as we have defined it, has the center point fixed and takes a test
function on
instead of a radius. The formula in the previous part defines a different sort of spherical mean of a
distribution :
What properties would you expect of this spherical mean applied to a harmonic distribution? What
are the advantages and disadvantages of this spherical mean compared to the one in the lecture script?
What interesting observations can you make about this?
(Bonus points as deserved.)
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26.
- Back in the saddle
Suppose that is a harmonic
function with a critical point at .
Assume that the Hessian of has
non-zero determinant. Show that
is a saddle point. Explain the connection to the maximum principle. (4 points)