Analysis of Partial Differential Equations
Exercise sheet III (Chapter 3)
1. (Sobolev embedding) Let Ω ⊂ Rd be a C 1 bounded domain. Prove that for all s ∈ N with s > d/2,
there exists a constant C > 0 such that
kukC s−d/2(Ω) ≤ CkukH s (Ω) .
(1)
Hint: Construct an ‘Extension Operator’ T : H 1 (Ω) → H 1 (Rd ) such that for all u ∈ H 1 (Ω), we have: (i)
T u|Ω = u, (ii) kT ukL2(Rd ) ≤ CkukL2 (Ω) , (iii) kT ukH 1 (Rd ) ≤ CkukH 1 (Ω) , where C depends only on Ω. In
order to construct the above extension operator, use the local charts to straighten (rectify) the boundary
and use the ‘partition of unity’. Once we have the extension operator, apply the result proved in lectures
for functions defined in Rd and reproduce all the arguments.
2. In the lectures, we have seen that W 1,p (Rd ) ⊂ L∞ (Rd ) for p > d. Give an example of a function
f ∈ H 1 (R2 ) but f 6∈ L∞ (R2 ).
3. (Poincaré-Wirtinger) Let Ω ∈ Rd be an open, bounded domain. Show that there exists a constant
C(Ω), depending on the domain, such that
Z
Z
2
1
|∇u|2 dx,
(2)
|u − ū| dx ≤ C(Ω)
∀ u ∈ H (Ω),
Ω
Ω
where ū =
1
|Ω|
Z
u(x) dx.
Ω
4. (Hardy) Let Ω ⊂ Rd be a C 1 bounded domain. Define the distance function d : Ω → R+ by
d(x) := dist(x, ∂Ω). Show that there exists a constant C > 0 such that
u
≤ Ck∇ukL2 (Ω) for all u ∈ H01 (Ω).
(3)
2
d L (Ω)
5. (Rellich) Let Ω ∈ Rd be an open and bounded domain. Prove that any sequence uniformly bounded
in H 1 (Ω) is relatively compact in L2 (Ω) i.e., if {un } ⊂ H 1 (Ω) is a sequence such that kun kH 1 (Ω) ≤ C
for some constant C independant of n, then there exists a subsequence {uϕ(n) } (with ϕ : N → N strictly
increasing) and a limit function u ∈ L2 (Ω) such that
uϕ(n) − u 2
→0
as n → ∞.
L (Ω)
6. (Riesz-Fréchet-Kolmogorov) Let Ω ∈ Rd be open.
• First consider ω ⊂⊂ Ω i.e., ω open with ω̄ ⊂ Ω. Consider a bounded G ⊂ L2 (Ω). Suppose
∀ ε > 0, ∃ δ > 0, δ < dist(ω, ∂Ω) such that
Z
|u(x + h) − u(x)|2 dx ≤ ε.
∀ h ∈ Rd , |h| < δ and ∀ u ∈ G,
(4)
ω
2
Prove that G|ω is relatively compact in L (ω). (Notation: G|ω denotes the elements of G restricted to ω).
1
• Second, assume that Ω = Rd and consider a bounded G ⊂ L2 (Rd ). Suppose
Z
∀ ε > 0, ∃ δ > 0 such that ∀ h ∈ Rd , |h| < δ and ∀ u ∈ G,
|u(x + h) − u(x)|2 dx ≤ ε.
(5)
Rd
In addition to (5), suppose also that
∀ ε > 0, ∃ ω ⊂⊂ Rd such that kukL2(Rd \ω) < ε ∀ u ∈ G.
(6)
Then, deduce that G is relatively compact in L2 (Rd ).
7. (Homogenization) Consider the following one dimensional Dirichlet problem:

∂ x ∂uε 
a
= f in (0, 1),
−
∂x
ε ∂x
 u (0) = u (1) = 0,
ε
(7)
ε
where a is a smooth, positive function which is 1-periodic and f ∈ L2 (0, 1).
(a) For any fixed ε > 0, prove existence and uniqueness of uε ∈ H01 (0, 1) for the Dirichlet problem (7).
(b) Consider a sequence of solutions {uε } associated with (7) for ε → 0. Show that uε ⇀ u0 in H01 (0, 1)
as ε → 0. Show also that the limit function u0 satisfies:

∂ ∂u0 
−
ā
= f in (0, 1),
(8)
∂x
∂x
 u (0) = u (1) = 0,
0
0
for some constant ā.
(c) How is ā represented in terms of the periodic function a?
Hint: Use Rellich Theorem and Riemann-Lebesgue Lemma.
8. (Fredholm Alternative) Let u ∈ H 1 (Ω) be a weak solution of the following Neumann problem:
(
b(x) · ∇u − ∇ · (A(x)∇u) = f in Ω,
−A(x)∇u · n = g
on ∂Ω.
(9)
where f ∈ L2 (Ω), g ∈ H 1 (Ω). The coefficient A(x) be a symmetric matrix such that there exist α0 > 0,
α1 > 0 with
α0 |ξ|2 ≤ Aij (x)ξi ξj ≤ α1 |ξ|2
for a.e. x ∈ Ω, ∀ ξ ∈ Rd .
(10)
and b(x) ∈ L∞ (Ω) satisfies ∇ · b = 0 in Ω and b · n = 0 on ∂Ω. Prove that (9) has a unique weak solution
modulo an additive constant if and only if the source terms satisfy the following compatibility condition:
Z
Z
f (x) dx =
g(x) dσ(x),
(11)
Ω
∂Ω
where dσ(x) is the surface measure on ∂Ω.
Hint: Employ Poincaré-Wirtinger inequality and use Lax-Milgram Lemma in the quotient space H 1 (Ω)/R.
9. (Mean Value Theorem) Let Ω ⊂ Rd be open. A function u ∈ C 2 (Ω) is said to be harmonic if ∆u = 0
in Ω. Suppose u is harmonic in Ω. Let x0 ∈ Ω and r > 0 so that the closed ball B̄(x0 , r) ⊂ Ω. Show that:
Z
1
u(x0 ) = d−1
u(y)dσ(y),
(12)
r
ωd S(x0 ,r)
where ωd is the surface area of the unit sphere in Rd and S(x0 , r) is the sphere of radius r centered at x0 .
10. (Liouville Theorem) Prove that every bounded harmonic function on the whole space Rd should be
a constant. Deduce that any harmonic function v(x) on the whole space Rd which satisfies |v(x)| →
0 as x → ∞ should be zero.
2
11. (Dirichlet Principle) Let Ω ∈ Rd be an open, bounded domain. For a source term f ∈ L2 (Ω), show
that solving for u ∈ H01 (Ω) satisfying
(
−∆u = f in Ω,
(13)
u=0
on ∂Ω,
is the same as solving for u ∈ H01 (Ω), the following minimization problem:
F (u) =
where
F (v) =
1
2
Z
inf
v∈H01 (Ω)
F (v),
|∇v|2 dx −
Ω
Z
(14)
f v dx.
Ω
12. (Helmholtz decomposition) Let Ω ⊂ Rd be an open and bounded domain. Suppose b(x) ∈ (L2 (Ω))d
is a vector field in Ω. Show that there exists u ∈ H01 (Ω) and v ∈ (L2 (Ω))d such that
b(x) = ∇u(x) + v(x)
with
∇ · v = 0 in Ω
and
Z
∇u · v dx = 0.
Ω
13. (Cacciopoli) Suppose Ω ⊂ Rd be open. Let x0 ∈ Ω and 0 < ρ < ρ̄ such that the ball B(x0 , ρ̄) ⊂ Ω.
Suppose u ∈ H 1 (Ω) satisfies
−∆u + b · ∇u + a u = 0 in Ω,
(15)
where a ∈ R and b ∈ Rd . Show that there exists a constant C such that
Z
Z
C
2
|u|2 dx.
|∇u| dx ≤
(ρ̄ − ρ)2 B(x0 ,ρ̄)
B(x0 ,ρ)
(16)
Take a = 0 and b = 0 in (15). Deduce from (16) that
∀ k ∈ N,
kuk2H k (B(x0 ,ρ)) ≤ C(ρ, ρ̄, k)kuk2L2(B(x0 ,ρ̄))
(17)
∀ k ∈ N,
kuk2C k (B(x0 ,ρ)) ≤ C(ρ, ρ̄, k)kuk2L2(B(x0 ,ρ̄)) .
(18)
and
What can we infer from (18)?
Hint: Use ‘Cut-off functions’ as was done in lectures for the interior regularity results.
14. (Maximum Principle - Divergence Form) Let A(x) be a symmetric matrix (i.e., Aij = Aji ) such that
there exist α0 > 0, α1 > 0 with
α0 |ξ|2 ≤ Aij (x)ξi ξj ≤ α1 |ξ|2
for a.e. x ∈ Ω, ∀ ξ ∈ Rd .
(19)
Also, let c(x) ∈ L∞ (Ω) and c(x) ≥ λ > 0. Suppose u ∈ H 1 (Ω) verifies in the weak sense:
−∇ · (A(x)∇u) + c u ≥ 0 on Ω,
which means in an explicit way that we have:
Z
Z
c uφ dx ≥ 0,
A(x)∇u · ∇φ dx +
Ω
Ω
∀ φ ∈ C0∞ (Ω) with φ ≥ 0 in Ω.
Show that
inf u(x) = inf u(x)
x∈Ω
x∈∂Ω
3
(20)
(21)
Hint:Use the density of C0∞ (Ω) in H01 (Ω). Take −(u − inf u(x))− as test function. We have used the
x∈∂Ω
following notation: h− := min(0, −h).
15. (Parabolic Equations) Let Ω ⊂ Rd be open. We
 ∂u


 ∂t + P u = f
u=0



u=g
consider the initial-boundary value problem (IBVP):
in Ω × (0, T ),
on ∂Ω × [0, T ],
(22)
on Ω × {t = 0},
where P is a second order partial differential operator in divergence form:
P u := −
d
d
X
∂ ∂u
∂u X
bi (t, x)
aij (t, x)
+
+ c(t, x)u.
∂xi
∂xj
∂xi
i,j=1
i=1
Define a time dependent bilinear form for u, v ∈ H01 (Ω) and for a.e. t ∈ [0, T ]:
B[u, v; t] :=
d Z
X
i,j=1
aij (t, x)
Ω
Z
d Z
X
∂u ∂v
∂u
dx +
bi (t, x)
vdx +
c(t, x)uvdx.
∂xj ∂xi
∂xi
Ω
i=1 Ω
(23)
We give the following definition for a weak solution of the IBVP (22): A function u ∈ L2 ((0, T ); H01 (Ω))
with u′ ∈ L2 ((0, T ); H −1 (Ω)) is a weak solution to (22) if
(i) hu′ , vi + B[u, v; t] = (f, v) for each v ∈ H01 (Ω) and for a.e. t ∈ [0, T ]
and (ii) u(0) = g,
(24)
where h·, ·i is the dual product between H −1 (Ω) and H01 (Ω) whereas (·, ·) is the standard L2 inner product.
The idea is to construct approximate solutions to (22) by considering an orthonormal basis {ϕk }∞
k=1 of
L2 (Ω). Define approximations for n ∈ N:
un (t) :=
n
X
dkn (t)ϕk ,
(25)
k=1
where the coefficient functions dkn : [0, T ] → R for k = 1, · · · , n are chosen such that:
dkn (0) = (g, ϕk ) for k = 1, · · · , n
and
(u′n , ϕk ) + B[un , ϕk ; t] = (f, ϕk ).
(26)
(15.a) Using an existence result from the theory of ODEs, show that for each n ∈ N there exists a unique
function un (t) of the form (25) satisfying (26).
Hint: Use the Cauchy-Lipschitz Theorem.
The next task is to consider the finite dimensional approximations un for n ∈ N and pass to the limit
as n → ∞. In order to apply some compactness results, we need to derive uniform (in n) estimates on
{un }. The next question addresses this aspect.
(15.b) There exists a constant C depending only on Ω, T, aij , bi , c such that
kun kL∞ ([0,T ];L2 (Ω)) + kun kL2 ((0,T );H01 (Ω)) + ku′n kL2 ((0,T );H −1 (Ω)) ≤ C kf kL2 ((0,T );L2 (Ω)) + kgkL2(Ω) . (27)
(15.c) Using the apriori estimates (27) and compactness results, arrive at the associated limit function
with the approximate solutions (25). Show that the limit function is indeed a weak solution of (22) in
the sense of (24) by passing to the limit in (26).
Hint: You may assume the following: If vn ⇀ v in L2 ((0, T ); H01 (Ω)) and vn′ ⇀ w in L2 ((0, T ); H −1 (Ω)),
then w = v ′ .
(15.d) Show that a weak solution of (22) is unique.
Hint: Assume that f = g = 0 in (22) and show that the solution u = 0.
4