total points: 20

Hand in y our solutions un til W ednesda y , 05/03/17, 14:15 (PO b o x of y our T A in V3-128)

total points: 20

Prof. Dr. Moritz Kaßmann Fakultät für Mathematik

Sommersemester 17 Universität Bielefeld

Partial Differential Equations Exercise sheet II, 04/26/17

Exercise II.1 (3+2 Points) Assume u(t, x) =v

x² t

for t >0and x∈R. (a) Prove thatu solves the equation

∂_tu−∂²_xu= 0 in(0,∞)×R if and only if forz≥0

4zv⁰⁰(z) + (2 +z)v⁰(z) = 0. (1) (b) Show that the general solution of (1) is

v(z) =c₁

z

Z

0

e^−s/4s^−1/2ds+c₂.

Exercise II.2 (5 Points)

Prove the following statement (Proposition 1.6): Letf ∈C(R^d)∩L¹(R^d). Then for every x₀ ∈R^d

Z

R^d

f(x)dx=

∞

Z

0





 Z

∂Br(x0)

fdO





 dr.

Furthermore, for everyr >0 andx0∈R^d

d ds

Z

Bs(x0)

f(x)dx s=r

= Z

∂Br(x0)

fdO.

Exercise II.3 (3+2 Points)

LetQ= (0,1)³andS={(x, y, z)∈R³|x²+y²+z² = 1}. Compute the following integrals with the help of the divergence theorem:

(a)

Z

Q

div(x², yz, y)^T dxdydz,

(b)

Z

S

3x²z²+y⁴+ 3y²x²+zy²x²dO.

Exercise II.4 (5 Points) Let

A={(x, y, z)∈R³|(x, y)∈B₁(0)⊂R²,0< z < h(x, y)}

for some positive function h∈C¹(B₁(0)).

Prove the divergence theorem for this specific choice ofA, i.e., forf ∈C¹(R³,R³) prove Z

A

divf = Z

∂A

(f, ν)dO.

2