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The Normal Contraction Property for Non-Bilinear Dirichlet Forms

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Abstract

We analyse the class of convex functionals \(\mathcal {E}\) over L2(X,m) for a measure space (X,m) introduced by Cipriani and Grillo (J. Reine Angew. Math. 562, 201–235 2003) and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if \(\mathcal {E}(\phi \circ f) \leq \mathcal {E}(f)\) for all f ∈L2(X,m), and all 1-Lipschitz functions \(\phi : \mathbb {R} \to \mathbb {R}\) 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 ∈L2(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.

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Acknowledgements

We are grateful to Giuseppe Savaré for numerous stimulating and helpful comments and to the anonymous referee for useful suggestions. We thank the organisers of the CEREMADE Young Researcher Winter School 2022, where part of the work was done.

Funding

The first author has been funded by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 754362. Partial support has been obtained from the EFI ANR-17-CE40-0030 Project of the French National Research Agency. The second author was supported by ERC Starting Grant 680275 “MALIG.”

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  1. CEREMADE, CNRS, Université Paris-Dauphine, PSL University, 75016, Paris, France

    Giovanni Brigati & Ivailo Hartarsky

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  1. Giovanni Brigati
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Correspondence to Giovanni Brigati.

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Brigati, G., Hartarsky, I. The Normal Contraction Property for Non-Bilinear Dirichlet Forms. Potential Anal 60, 473–488 (2024). https://doi.org/10.1007/s11118-022-10057-2

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