The lack of strict convexity and the validity of the Comparison Principle for a simple class of minimizers

Let w*(x) = a . x + b be an affine function in R(N), Omega subset of R(N), L : R(N) -> R be convex and w be a local minimizer of I(v) = integral(Omega) L(del v(x))dx in W(1,1)(Omega, R) with w(x) <= w*(x) on partial derivative Omega in the trace sense. Then w* satisfies the Comparison Principle from above, i.e. w(x) <= w*(x) a.e. on Omega if and only if (a, L(a)) does not belong to the relative interior of a N-dimensional face of the epigraph of L. As a consequence, if F is the projection of a bounded face of the epigraph of L, the local minimizer w*(x) = max{xi . (x - x(0)) : xi is an element of F} satisfies the Comparison Principle from above if and only if dim F <= N - 1 or x(0) is not an element of Omega. (C) 2010 Elsevier Ltd. All rights reserved.