The normal contraction property for non-bilinear Dirichlet forms

Abstract

We analyse the class of convex functionals $\mathcal E$ over $\mathrm{L}^2(X,m)$ for a measure space $(X,m)$ introduced by Cipriani and Grillo and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if $\mathcal E(φ\circ f) \leq \mathcal E(f)$ for all $f \in \mathrm{L}^2(X,m)$, and all 1-Lipschitz functions $φ: \mathbb R \to \mathbb R$ with $φ(0)=0$. We prove that normal contraction holds if and only if $\mathcal E$ is symmetric in the sense $\mathcal E(-f) = \mathcal E(f),$ for all $f \in \mathrm{L}^2(X,m).$ 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 $φ$.

Figures (8)

• Figure 1: Graph of the function $H_{\alpha}(f,g)(x)$ for fixed $g(x)$.
• Figure 2: Graph of the function $\phi_{x_1,x_2,x_3}$.
• Figure 3: Illustration of \ref{['eq:minmax4']}.
• Figure 4: Illustration of \ref{['eq:HK5']}.
• Figure 5: Illustration of \ref{['eq:HK4']}.

Theorems & Definitions (20)

• Example 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 1.4
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 2.4
• proof : Proof of \ref{['thm1']}
• Proposition 3.1