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

Let w∗(x)=a⋅x+bw∗(x)=a⋅x+b be an affine function in RNRN, Ω⊂RNΩ⊂RN, L:RN→RL:RN→R be convex and ww be a local minimizer of View the MathML sourceI(v)=∫ΩL(∇v(x))dx Turn MathJax on in W1,1(Ω,R)W1,1(Ω,R) with w(x)≤w∗(x)w(x)≤w∗(x) on ∂Ω∂Ω in the trace sense. Then w∗w∗ satisfies the Comparison Principle from above, i.e. w(x)≤w∗(x)w(x)≤w∗(x) a.e. on ΩΩ if and only if (a,L(a))(a,L(a)) does not belong to the relative interior of a NN-dimensional face of the epigraph of LL. As a consequence, if FF is the projection of a bounded face of the epigraph of LL, the local minimizer View the MathML sourcew∗(x)=max{ξ⋅(x−x0):ξ∈F} satisfies the Comparison Principle from above if and only if dimF≤N−1dimF≤N−1 or x0∉Ωx0∉Ω.