EXERCISE SET 13
PARTIAL DIFFERENTIAL EQUATIONS 2, 2017
DEADLINE 27/04/2017 at 12h15
1.BM O space
Show that L2,n (Q0 ) ⊂ BM O(Q0 ).
2.A case of the Ehrling-Gagliardo-Nirenberg theorem
Use integration by parts to show that
Z
2
Z
|Du| dx ≤ C
Ω
1/2 Z
2
2
1/2
|D u| dx
u dx
Ω
2
Ω
for all u ∈ Cc∞ (Ω). Assume ∂Ω is smooth, and prove this inequality if u ∈ W 2,2 (Ω)∩W01,2 (Ω).
(Hint: an approximating smooth sequence uk )
3.Corolary of Hardy-Littlewood maximal theorem
Show that if f ∈ L∞ (Q0 ) then M f ∈ L∞ (Q0 ) and ||M f ||L∞ (Q0 ) ≤ ||f ||L∞ (Q0 ) .
Using the previous result and the Hardy-Littlewood maximal theorem, prove that there
exists a constant such that
||M f ||Lp (Q0 ) ≤ C ||f ||Lp (Q0 )
∀f ∈ Lp (Q0 ).
4.Inclusion
Show that W 1,n (Q0 ) ⊂ BM O(Q0 )
5.Properties of sharp functions
Prove the proposition in the lecture notes, that is
i) f ∈ BM O(Q0 ) if and only if f ] ∈ L∞ (Q0 ).
ii) If 1 < p < ∞ and f ∈ Lp (Q0 ) then f ] ∈ Lp (Q0 ) and
] f p
≤ C(n, p) ||f ||Lp (Q0 ) .
L (Q0 )
6. (2 points)Counterexample for f continuous
a)Let f a function on B1 ⊂ R2 defined by f (0) = 0 and for x ∈ B1 \ {0}
2
1
x22 − x21
.
+
f (x) =
|x|2
(− ln(|x|))1/2 4(− ln(|x|))3/2
1
Show that f is continuous and that the function u defined on B1 by u(0) = 0 and for x 6= 0
u(x) = (x21 − x22 )(− ln(|x|))1/2
satisfies u ∈ C(B1 ) ∩ C ∞ (B1 \ {0}) and
in B1 \ {0}
∆u = f
but u ∈
/ C 2 (B1 ).
b)We have the following theorem (admitted):
Theorem 1. Suppose that w is harmonic in BR \ {0} ⊂ Rn , that is u ∈ C 2 (BR \ {0}) and
∆w = 0 in BR \ {0}.
If w is continuous at 0 or if w satisfies
o(ln(|x|) if n = 2
w(x) =
as |x| → 0.
o(|x|2−n ) if n > 2
Then w can be defined at 0 so that is harmonic in BR .
Using the following Theorem on removability of isolated singularities for harmonic maps,
show that the problem
∆h = f
in B1
does not have a C 2 solution.
Hint: Suppose that there exists a C 2 solution v and consider the difference with u. What
can we say about v − u in BR for R ∈ (0, 1)? what can we say then about u? conclude.
Bonus. 2 points
Show that if u ∈ BM O(Q0 ) then u+ ∈ BM O(Q0 ) and |u| ∈ BM O(Q0 ).
Show that if f ∈ L∞ then solutions to −∆u = f have a gradient which are locally Hölder
continuous. (use easy arguments).
2