Partial Differential Equations
MATH 612
A. Bonito
Due April 9
Spring 2013
Assignement 3
Exercise 1
30%
We saw in class that for two Hilbert spaces X and Y and a continuous bilinear form b : X × Y → R
the inf sup condition : there exists β > 0 such that
inf
sup
q∈Y, kqkY =1 v∈X, kvkX =1
b(v, q) > β
implies that the operatore B : X0⊥ → Y ∗ is an isomorphism. Here
X0 := {v ∈ X : b(v, q) = 0,
∀q ∈ Y } .
Prove that if B is an isomorphism such that there exists a constant C satisfying
kBϕkY ∗ 6 CkϕkX
for all ϕ ∈ X0⊥ , then the inf-sup condition holds.
Exercise 2
35%
Let Ω ⊂ R2 be open, bounded and with smooth boundary. Given f ∈ L2 (Ω)2 , we consider the
stationary Stokes system
−∆u + ∇p = f,
and
divu = 0,
in Ω
supplemented by the boundary condition u = 0 on ∂Ω.
Show that there exists a unique solution (u, p) ∈ H01 (Ω)2 × L20 (Ω) where
Z
2
2
L0 (Ω) := q ∈ L (Ω) :
q=0 .
Ω
Hint : You may assume that there exists a constant C such that for every q ∈ L20 (Ω), there exists
a v ∈ H01 (Ω)2 such that
divv = q
with
kvkH 1 (Ω)2 6 CkqkL2 (Ω) .
Exercise 3
35%
Let Ω ⊂ R2 be open, bounded and with smooth boundary. Given f ∈ L2 (Ω), we consider the mixed
formulation of the Poisson equation
u = ∇p,
and
− divu = f,
in
Ω
supplemented by the boundary condition p = 0 on ∂Ω.
Find the adequate spaces X and Y such that the above system has a unique solution (u, p) ∈ X ×Y .
Justify your answer by proving that with the proposed choice, there exists, indeed, a unique
solution.