Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
842116 | Nonlinear Analysis: Theory, Methods & Applications | 2010 | 7 Pages |
Abstract
Let wâ(x)=aâ
x+b be an affine function in RN, ΩâRN, L:RNâR be convex and w be a local minimizer of I(v)=â«Î©L(âv(x))dx in W1,1(Ω,R) with w(x)â¤wâ(x) on âΩ in the trace sense. Then wâ satisfies the Comparison Principle from above, i.e. w(x)â¤wâ(x) a.e. on Ω 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 wâ(x)=max{ξâ
(xâx0):ξâF} satisfies the Comparison Principle from above if and only if dimFâ¤Nâ1 or x0âΩ.
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Authors
Carlo Mariconda,