AbstractRecently N. Korevaar developed a method of proving that solutions to elliptic and parabolic boundary value problems on convex domains ω ⊂Rnare convex functions. He introduced a concavity functionand used the classical maximum principle to prove thatC⩾ 0 on ω × ω, i.e. thatuis convex. Both he and independently L. Caffarelli and J. Spruck applied this method successfully to various boundary value problems. In this note we weaken the assumptions of their theorems and obtain some interesting new applications which are not covered by their previous results [