1. Show that if A and B are positive-semidefinite matrices then
X
aij bij ≥ 0.
i,j
More generally, show that A · B is positive-semidefinite, where A · B = (aij bij )i,j is the Schur (or
Hadamard) product of matrices.
2. Assume a domain U is locally defined by the condition γ(x) < 0 for a C 2 function γ. Show
that U satisfies the interior ball condition at every boundary point x0 such that Dγ(x0 ) 6= 0. Show
that if Dγ(x0 ) = 0 then the interior ball condition is not necessarily satisfied.
3. Show that if
L=−
X
ij
is elliptic and u ∈
C2
aij
X ∂
∂2
+
bi
∂xi ∂xj
∂xi
i
is such that Lu = 0 then for any smooth convex function ϕ we have Lϕ(u) ≤ 0.
1