The normal contraction property for non-bilinear Dirichlet forms

Giovanni Maria Brigati (1) , Ivailo Hartarsky

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Primary 31C45; Secondary 47H20; 31C25; 46E36; 35K55 Non-bilinear Dirichlet form; Dirichlet form; nonlinear semigroup; gradient flow; normal contraction; Cheeger's energy

Domaines: Mathématiques [math]

Giovanni Maria Brigati, Ivailo Hartarsky. The normal contraction property for non-bilinear Dirichlet forms. 2022. ⟨hal-03663291⟩