Math 295 Winter Term 2008
Assignment 7
1. For α∈(0,1), 0 ≤s0< s1and s= (1 −α)s0+αs1prove the
interpolation inequality
kukHs≤ckuk1−α
Hs0kukα
Hs1, u ∈Hs1
first for Rnand then for a bounded domain with smooth boundary.
2. Let Banach spaces Ej,j= 0,1,2, be given with
E2,−,→E1,→E0.
Show that, given ε > 0, there is a constant cε>0 such that
kukE1≤εkukE2+cεkukE0, u ∈E2.
3. Show that the trace operator γ∂Hnsatisfies
γ∂HnH2(Rn)= H3/2(∂Hn).
4. Let Ω ⊂Rnbe a bounded domain with smooth boundary, α > 0
and f∈L2(Ω). Find the boundary value problem for which
ZΩ
∇u· ∇v dx +ZΩ
uv dx +αZ∂Ω
u v dσ∂Ω=ZΩ
fv dx ∀v∈H1(Ω)
is the appropriate weak formulation and show that it possesses a
unique solution.
5. Let Ω ⊂Rnbe a bounded domain with smooth boundary and find
the natural weak formulation of the bvp
42u=f∈L2(Ω) , u = 0 , ∂νu= 0 on ∂Ω
and prove that it has a unique solution.
Homework due by Wednesday, January 23 2008
