Due Friday, Feb 16 at noon (under my office door).
Office hours: Monday 12-2pm.
Ch 1 (pp12-13)
4. Taylor's thm in multi-index notation.
Ch 2 (pp85-89)
2. Laplace eqn invariant under rotations.
X1. ((a) is quite easy! :-) )
4. Subharmonic functions
5. Ball bound.
X2. Let U = (0,1) in R1, S = {u:U->R | u non-negative, u harmonic, u not identically 0},
V = (0.1,0.7). What is supu in S ( supV u / infV u )?
X3. Find the Green's function for the region U in R3, U={(x,y,z), 0 < z < 1}. Consider
an infinite sequence of "image" sources.
Office hours: Mondays 12-2pm.
Ch 2 (pp85-89)
X1. Show that the backwards heat equation initial (final) value
problem is ill-posed because the solution does not depend
continuously on the initial (final) conditions. Hint: consider
perturbations of the t=0 conditions which are of the form (1/k)cos(kx),
and show that for any &tau < 0, there are changes of arbitrarily
small size (L-infinity) to u(x,0) that correspond to changes to
u(x,&tau) of arbitrarily large size. (Consider k->infinity.)
X2. (a) Verify that the alleged solution given in class Feb 15
of the heat equation Cauchy problem u(x,0)=0 does formally solve
the heat equation (x in R1) for all t>0.
(b) Make a sketch of graph of
the solution. (Note: 2k ! means (2k)!.)
Note: I am asking this beacuse I am curious to know what this very surprising
solution of the heat equation looks like.
I have read that the series converges for all x in R, t>0, and has limit 0 for all x
as t approaches 0 from above. But I have not seen a proof of this or a picture of
the solution. It would be nice to have at least the latter.
X3. Step 3 of the proof of Thm 8 (smoothness of solutions of HE)
seems a little sketchy or incomplete. Provide and prove a proposition of the form that if K
is smooth (define domain carefully) and u(x,t) = some integral K(x,t,y,s) u(y,s) dyds,
for (x,t) in some region, then u is smooth.
X4. Find a solution of the heat equation in 1 spatial dimension
on a "UT"
that gives as large a value as possible for the ratio
max on C(x,t;r/2) of rn+3 times | ux | divided
by the L1 norm on C(x,t;r) of u for some cylinder C(x,t;r)
contained in UT.
Specify the cylinder, your solution and the value of your ratio.
More points given for solutions with higher ratios.
X5. Prove Lemma 2(i) on p75.
X6. Confirm that u defined by (31) satisfies part (iii) of Thm 2, p77.
2.5.16 (Maxwell's equations)
X7. 
X8. 
X1. 
X2. In the Burgers equation with g(x)=x3 - 3 x + 3,
where and when does the "shock" first appear? (Show all your work.)
3.5.1 (complete integral of HJ eqn)
3.5.2