Problems
Problems from the text:
page 12, .problem 3
page 290 problems 5,6,7,8
page 291 problems 10,13,14,16,17
page 292 problem 18
page 345 1,2,3,4,7
page 425 1 through 10 (Hint for 2 and 6: expand u(x,t) in terms of an appropriate family of
eigenfunctions ád n ÝxÞâÞ
1. (equivalence of definition of weak derivative)
Show that for u 5 L 2 ÝR 1 Þ, the following are equivalent:
(a) / x u 5 L 2 ÝR 1 Þ
(b) z û 5 L 2 ÝR 1 Þ
uÝx + hÞ ? uÝxÞ
(c)
converges in L 2 ÝR 1 Þ as h ¸ 0
h
(d) there exists a sequence of test functions d n , such that d n converges to f
in L 2 ÝR 1 Þ and / x d n converges in L 2 ÝR 1 Þ
2. For what values of s is the characteristic function of I = ß0, 1à in H s ÝRÞ?
For what values of s is the characteristic function of I 2 = ß0, 1à 2 in H s ÝR 2 Þ?
3. To which spaces H m ÝUÞ do the following functions u belong?
ÝaÞ
uÝxÞ =
0
if 0 < x < 1
x?1
if 1 ² x < 2
x 3 ? x 2 ? 3 if 2 ² x < 3
ÝbÞ uÝx, yÞ =
xy
0 < x < 1, 0 < y < 1
xÝ2 ? yÞ
0 < x < 1, 1 < y < 2
0 < x < .5, y = 1
x
4
x
x = .5, y = 1
.5 < x < 1, y = 1
1
4) Let
uÝxÞ =
if 0 < x < 1,
x
and vÝxÞ = Sin^x
2 ? x if 1 < x < 2,
a) Determine whether u and v are orthogonal in L 2 Ý0, 2Þ.
Are they orthogonal in H 1 Ý0, 2Þ?
b) Find the distance from u to v in L 2 Ý0, 2Þ and in H 1 Ý0, 2Þ.
5. Use the Sobolev embedding theorem to show that the function
uÝx, yÞ =
x 2 y 2 if x > 0, y > 0
0
otherwise
is continuous on I = Ý?1, 1Þ × Ý?1, 1Þ
6. Suppose u 5 H 4 ÝUÞ, and v 5 H 2 ÝUÞ . Show that
XU 4 2 u4 2 vdx = XÝ4 4 uÞvdx + X/U ßÝ4 2 uÞ/ N v ? / N Ý4 2 uÞvàdS
Here
/N = n 6 4
where n = outward unit normal to /U
7. By evaluating the relevant integrals, show that:
x and x ln x 5 H 1 Ý0, 1Þ,
ln x 5 L 2 Ý0, 1Þ but x ?1 6 L 2 Ý0, 1Þ
x ?1 5 H ?1 Ý0, 1Þ Ýa function belongs to 5 H ?1 Ý0, 1Þ if it is the weak derivative of an
element of L 2 Ý0, 1ÞÞ
8. Suppose uÝxÞ is continuous for all x in R. Show that for all test functions dÝxÞ,
lim h¸0 X dÝxÞßuÝx + hÞ ? uÝxÞà/hdx = ? X uÝxÞd v ÝxÞdx
9. Let U = Ýa, bÞ denote a bounded open set in R.
a) Show that if u 5 H 10 ÝUÞ, then u is absolutely continuous and uÝaÞ = uÝbÞ = 0.
b) Show that H 1 ÝUÞ is the direct sum of H 10 ÝUÞ and the space of functions
vÝxÞ = Ae x + Be ?x for arbitrary constants A, B; i..e, show that any v of this
form is orthogonal to every function in H 10 ÝUÞ.
c) Show that for every u in C K0 ÝUÞ,
x
uÝxÞ 2 = X 2u v ÝzÞuÝzÞdz
a
and then use the C-S inequality to show that for some constant C > 0,
2
b
b
X a uÝzÞ 2 dz ² C X a u v ÝzÞ 2 dz
d) Show that the previous inequality continues to hold for all u in H 10 ÝUÞ.
10. Let
u 1 ÝxÞ =
u 2 ÝxÞ =
1 ? |x|
if | x| < 1
0
if |x| > 1
x 2 if |x| < 1,
0 if |x| > 1
Find u v1 ÝxÞ, the weak derivative of u 1 ÝxÞ and show that it is a locally integrable function.
Show that the weak derivative of u 2 ÝxÞ is not locally integrable
11. Show that for a bounded linear operator A on H, if qAxq H ³ cqxq H for all x in H,
then the range of A is a closed subset of H.
12. Suppose uÝx, yÞ 5 H m ÝR 2+ Þ and let
uÝx, yÞ
vÝx, yÞ =
if
y>0
m
> a k uÝx, ?kyÞ if y < 0
k=1
where
m
>Ý?kÞ s a k = 1 for 0 ² s ² m ? 1.
k=1
Show that
/ px / qy vÝx, yÞ
/ px / qy uÝx, yÞ
=
if
y>0
m
>Ý?kÞ q a k / px / qy uÝx, ?kyÞ if y < 0
k=1
and that this implies
lim y¸0 ? / qy vÝx, yÞ = lim y¸0+ / qy uÝx, yÞ
for 0 ² q ² m ? 1.
Does this imply that the mapping defined by vÝx, yÞ = EuÝx, yÞ maps H m ÝR 2+ Þ continuously
into H m ÝR 2 Þ?
3