AM 127: Solution to Homework 5
Problem 1. Let Rn be a bounded open set with Lipschitz boundary and h L2 ().
Show that
1
1
| u(x)|2 h(x)u(x) dx : u W0 ,2 ()
2
inf I (u) =
=m
1
has a unique solution u W0 ,2 () which satises in addition
1
h(x)(x)dx, W0 ,2 ()

AM/ACM 127 - HW4 Solutions
Problem 1.
Let
A1 = cfw_x R3 : |x| > 1,
A2 = cfw_x R3 : |x| < 1,
and consider
f (x) = |x|
for R.
(i) For which values of p and is f Lp (A1 ), or f Lp (A2 )?
(ii) For which k, p, is f in W k,p (A1 ) or in W k,p (A2 )?
Solution
(i

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t
f.H
\
- l" l
= \ Pt-*i
i
,H
s onsrvico,l t ,1*r.li,.r* ,
\
Y n = .f f
cfw_e- Jt
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o
P
It
an2ularr
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c
tf* canorri al *.1wot,
^a
d
U5 q-
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w'n- havr :
uln$e.
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g16

AM 127: Final Exam
Due: 4 pm, Thursday, June 6, 2013
Rules
This exam is timed: 3 hours in one sitting.
Do not open this exam until you are ready to take it.
You can use the textbook (Jost and Li-Jost), instructors notes, your class notes, your
homework