PROBLEM SET 5, 18.155
DUE 16 OCTOBER, 2015
There is only one lecture this week and I decided to keep this problem set short – it is mostly based on the Lecture on Thursday Oct 15. After
Q6 the questions are for your amusement
Q1. Show that if m < n then for a homogeneous elliptic polynomial,
P ( ξ ) = P m
( ξ ) , 1 /P m
( ξ ) ∈ S
0
(
R n
) is well-defined and P m
( D ) has a homogeneous fundamental solution.
Q2. For the case of ∂ = ∂ x
+ i∂ solution is E = c/ ( x + iy ) .
y in
R
2 show that the fundamental
Hint: Show that if E is the homogeneous fundamental solution then ( x + iy ) E ∈ S (
R n ) is in the null space and is homogeneous of degree 0 .
Q3. Show that any ‘holomorphic function’, meaning an element of u ∈ C −∞
(Ω) for Ω ⊂
R
2 , satisfying ∂u = 0 is smooth (Easy, just quote).
Q4. Suppose u ∈ C −∞
(Ω) , Ω ⊂
R
2 open and ∂u = 0 in Ω .
Suppose
B = {| z − ¯ | < } is a ball inside this open set. Show that
(1) ∂ ( χ
B u ) may be written as a contour integral around the boundary of
B (do it for a more general smoothly bounded domain if you have the enthusiasm).
If you just assume u ∈ C ∞
(Ω) you need to add a second ‘bulk’ talk to the right side, namely ( ∂u ) χ
B
.
Q5. With E the fundamenatal solution of ∂ above, show that
E ∗ ∂ ( χ
B u ) gives Cauchy’s integral formula for u inside B if ∂u = 0 .
Q6. Deduce that any solution of ∂u is analytic in the sense that it is locally the sum of a convergent power series in z near each point – namely its Taylor series.
Q7. Deduce the ‘Uniqueness of analytic continuation’: If Ω ⊂
R
2 is open and connected then two analytic functions in Ω which have the same Taylor series at one point are equal throughout
Ω .
1

2 PROBLEM SET 5, 18.155 DUE 16 OCTOBER, 2015
Hint: First do this for Ω an open ball with the point being the center – i.e. if an analytic function has Taylor series vanishing at the center of an open ball it vanishes in the ball. This follows by using Q6 above to show the function vanishes in some open ball around the center and then take the largest such, it it is not the whole ball then apply the same argument around each point in its boundary to arrive at a contradiction. In the general case you can now see that the support of the difference must be open as well as closed.
Q8. A map u : Ω −→ B from an open set in
R
2 into a Banach space is weakly analytic if f ( u ) is holomorphic for each f ∈ B
0
.
It is strongly analytc if it is the sum, in B, of a power series in z near each point of Ω .
Show that these are equivalent conditions.
and their duals. Is it true?