Math 6342 Homework 3
Due date: Mar 7 (Tue)
Note: Some hints are given in a separate file. For best results, try to refrain from looking at them
unless you have been stuck for too long :)
1. For the boundary H 2 estimate in Theorem 4 of Sec 6.3.2, show that it is necessary to have
the assumption that u has trace zero. That is, show by example that there is no estimate
kukH 2 (U ) ≤ C(kf kL2 (U ) + kukL2 (U ) )
if the trace zero assumption is removed.
2. Evans, p369, #8. (If you are using the first edition, it’s #5 in page 346.)
P
3. (maximum principle for weak solutions) Let Lu = − ni,j=1 (aij (x)uxi )xj + c(x)u, where L is
uniformly elliptic, aij , c ∈ C(Ω̄), aij = aji , and c ≥ 0. Let u ∈ H 1 (Ω) ∩ C(Ω̄) be a weak
subsolution of Lu = 0 in Ω. By weak subsolution, we mean that


Z
n
X

aij (x)uxi vxj + c(x)uv  dx ≤ 0
Ω
i,j=1
for all v ∈ H01 (Ω), v ≥ 0. Show that
max u ≤ max u+ ,
∂U
Ω̄
where u+ := max{u, 0}.
4. Let Ω be a bounded domain in Rn with smooth boundary. If u ∈ C 2 (Ω) ∩ C(Ω̄) satisfies
∆u = u3 − u in Ω and u = 0 on ∂Ω, show that −1 ≤ u(x) ≤ 1 for all x ∈ Ω. Is it possible to
have u(x0 ) = 1 or −1 for x0 ∈ Ω?
0. Optional bonus question: The regularity results and maximum principle for linear elliptic
equations can be helpful to prove existence of solution for certain nonlinear equations. Let Ω =
B(0, 1) be the unit ball in Rn . Prove that there exists a classical solution u ∈ C 2 (Ω) ∩ C(Ω̄)
to the boundary value problem
(
−∆u = arctan(u + 1) in Ω,
u=0
on ∂Ω.
1