Fachbereich Mathematik & Informatik
Freie Universität Berlin
Prof. Dr. Carsten Gräser, Tobias Kies
3rd exercise for the lecture
Numerics III
Summer Term 2016
http://numerik.mi.fu-berlin.de/wiki/SS_2016/NumericsIII.php
Due: Wednesday, May 18th, 2016 (10:15 a.m.)
Exercise 1 (8 TP)
Let Ω ⊆ Rn with smooth boundary.
Furthermore, suppose that γ : Rn → R is strictly
R
convex and define J(u) := Ω γ(∇u(x)) dx for all u ∈ C 1 (Ω).
a) For each f ∈ C 1 (Ω) define Vf := {u ∈ C 1 (Ω) | u|∂Ω = f |∂Ω }. Show that J is
strictly convex on the set Vf for any f ∈ C 1 (Ω).
b) Show that there can be at most one u∗ ∈ Vf such that J(u∗ ) = inf u∈Vf J(u).
c) From now on assume that the minimizer u∗ ∈ Vf of J over Vf exists. Show that
Z
∇γ(∇u∗ (x)) · ∇v(x) dx = 0.
∀v ∈ V0 :
Ω
d) Assume furthermore that γ(x) =
Prove that for every x ∈ Ω
div
p
1 + kxk2 for all x ∈ Rn and that u∗ ∈ C 2 (Ω).
∇u∗ (x)
p
1 + k∇u∗ (x)k2
!
= 0.
Exercise 2 (8 TP)
a) Suppose a map A ∈ C(Rn , Rn×n ), a vector field b ∈ C(Rn , Rn ) and a scalar function
c ∈ C(Rn , R). The differential operator defined by
L(x, u) := A(x) : D2 u(x) + b(x) · ∇u(x) + c(x)u(x)
is called elliptic in x ∈ Rn if and only if all eigenvalues of A(x) are unequal 0 and
have the same sign.
Let n = 2. Show that L is elliptic in x ∈ Rn in the sense of (a) if and only if
A11 (x)A22 (x) − A12 (x)A21 (x) > 0.
Please turn over...
b) Let m ∈ N and aα ∈ C(Rn , R) for every multi-index α with |α| ≤ m. The
differential operator defined by
X
aα (x)∂α u(x)
L(x, u) :=
|α|≤m
is called elliptic in x ∈ Rn if and only if for all ξ ∈ Rn \ {0}
X
aα (x)ξ α 6= 0.
L0 (x, ξ) :=
|α|=m
Show that every elliptic differential operator in the sense of (a) is elliptic in the
sense of (b).
c) Suppose that L is an elliptic differential operator in the sense of (b). Prove that
m is even.
Remarks:
– For two matrices A, B ∈ Rn×n we define A : B :=
the element-wise products of A and B.
Pn
i,j=1 Aij Bij
to be the sum of
– An operator in the sense of (a) is called parabolic if A has one zero-eigenvalue
and all other eigenvalues are unequal 0 and have the same sign. The operator L
is called hyperbolic if A has one strictly negative (or positive) eigenvalue and all
other eigenvalues are strictly positive (or negative).
Q
– For a vector ξ ∈ Rn and a multi-index α = (α1 , . . . , αn ) we define ξ α := ni=1 ξiαi .
– The last statement is even true for complex coefficients if n ≥ 3.
Exercise 3 (4 extra-TP)
Classify the following PDEs.
a) −div(α(x)∇u) = 0,
~
b) ε∆u − β∇u
= 0,
α ∈ C 1 (Ω),
(i) for ε 6= 0,
α(x) ≥ α0 > 0.
(ii) for ε = 0.
c) (c20 − u2x )uxx − 2ux uy uxy + (c20 − u2y )uyy = 0
Have fun!
c0 > 0.