论文标题
非双线性迪里奇的正常收缩特性
The normal contraction property for non-bilinear Dirichlet forms
论文作者
论文摘要
我们分析了$ \ mathrm {l}^2（x，m）$ over $ \ mathcal e $ $ \ mathcal e $的类别$ \ mathcal e $（x，m）$，用于cipriani和grillo引入的测量空间$（x，m）$，并概括了经典的双线性dirichlet表单。我们研究了这种非线性形式是否验证了正常的收缩属性，即，如果$ \ natcal e（ϕ \ circ f）\ leq \ leq \ leq \ mathcal e（f）$（f）$ in \ in \ in \ mathrm {l}^2（l}^2（x，m）$ $ ϕ（0）= 0 $。我们证明，正常的收缩才能达到，并且仅当$ \ nathcal e $从某种意义上是对称的，$ \ mathcal e（-f）= \ Mathcal e（f），所有$ f \ in \ mathrm {l}^2（x，m）中的所有$ f \ $ f \ in n \ mathrm {l}^2（x，m）。 $ ϕ $。
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 $ϕ$.
