UNIVERSITY OF OSLO
Faculty of Mathematics and Natural Sciences
Examination in
MAT-INF4300 — Partial differential
equations and Sobolev spaces I.
Day of examination: Thursday, December 10, 2009.
Examination hours:
14.30 – 17.30.
This problem set consists of 2 pages.
Appendices:
None.
Permitted aids:
Approved calculator.
Please make sure that your copy of the problem set is
complete before you attempt to answer anything.
Problem 1
a. If u and v are in H 1 (R), show that
Z
Z
0
uv dx = − u0 v dx.
R
R
b. If u and v are in H 1 (R), show that the product uv also is in H 1 (R).
(Hint: Recall that H 1 (R) ⊂ C 0,1/2 (R).)
Problem 2
Assume that U is a bounded open subset of Rn with a C 1 boundary. Let
Z
1
X = u ∈ H (U )
u(x) dx = 0 .
U
a. Show that there is a constant C such that
Z
Z
2
u dx ≤ C
|Du|2 dx for all u ∈ X.
U
U
b. Show that there is no constant C such that
Z
Z
2
u dx ≤ C
|Du|2 dx for all u ∈ H 1 (Rn ).
Rn
(Continued on page 2.)
Rn
Examination in MAT-INF4300, Thursday, December 10, 2009.
Page 2
Problem 3
Let U and X be as the previous exercise. Consider the differential equation
(
−∆u = f in U ,
(1)
∂u
= 0 on ∂U ,
∂ν
where ν is the unit normal on ∂U .
a. If u ∈ C 2 (U ) ∩ C 1 (Ū ), is a (classical) solution to (1) show that
Z
f dx = 0.
(2)
U
b. Show that for each f ∈ L2 (U ) there is a unique u ∈ X such that
Z
Z
Du · Dv dx =
f v dx for all v ∈ X.
U
c. If f ∈ L2 (U ), but
U
R
U
f dx 6= 0, (3) still gives a solution u. Explain.
END
(3)