The normal contraction property for non-bilinear Dirichlet forms

Abstract

We analyse the class of convex functionals E over L 2 (X, m) for a measure space (X, m) introduced by Cipriani and Grillo [17] and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if E(φ • f ) E(f ) for all f ∈ L 2 (X, m), and all 1-Lipschitz functions φ : R → R with φ(0) = 0. We prove that normal contraction holds if and only if E is symmetric in the sense E(−f ) = E(f ), for all f ∈ L 2 (X, m). An auxiliary result, which may be of independent interest, states that it suffices to establish the normal contraction property only for a simple two-parameter family of functions φ.