The normal contraction property for non-bilinear Dirichlet forms
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913214074191872 |
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| author | Brigati, Giovanni Hartarsky, Ivailo |
| author_facet | Brigati, Giovanni Hartarsky, Ivailo |
| contents | 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 $ϕ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02928 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The normal contraction property for non-bilinear Dirichlet forms Brigati, Giovanni Hartarsky, Ivailo Functional Analysis Analysis of PDEs 31C45 (Primary), 47H20, 31C25, 46E36, 35K55 (Secondary) 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 $ϕ$. |
| title | The normal contraction property for non-bilinear Dirichlet forms |
| topic | Functional Analysis Analysis of PDEs 31C45 (Primary), 47H20, 31C25, 46E36, 35K55 (Secondary) |
| url | https://arxiv.org/abs/2205.02928 |