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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