Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

Claus, Burkhard

doi:10.1007/s11118-021-09935-y

Cited by 4 publications

(3 citation statements)

References 13 publications

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“…Then, the space D(E) is called Dirichlet space. Some properties of D(E) have already been given in [20], but, under our hypotheses, we have a simpler structure with new results.…”

Section: Energy Spacesmentioning

confidence: 85%

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The Normal Contraction Property for Non-Bilinear Dirichlet Forms

Brigati

Hartarsky

2022

Potential Anal

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“…Then, the space D(E) is called Dirichlet space. Some properties of D(E) have already been given in [20], but, under our hypotheses, we have a simpler structure with new results.…”

Section: Energy Spacesmentioning

confidence: 85%

“…• A definition of nonlinear Dirichlet form has been given in [19]. Properties of nonlinear Dirichlet forms have been studied in [21,20] and [17,16]. No analogous of the Bakry-Emery Γ-calculus is available in this context up to the best of our knowledge.…”

Section: Introductionmentioning

confidence: 99%

The Normal Contraction Property for Non-Bilinear Dirichlet Forms

Brigati

Hartarsky

2022

Potential Anal

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“…In this note, we use an abstract compactification Ω of Ω, an abstract boundary of Ω, and a capacity on Ω, which were recently considered in the case of nonlinear Dirichlet forms by Claus [7]. Let us explain this in more detail.…”

Section: Notation Andmentioning

confidence: 99%

Domination of semigroups generated by regular forms

Arora¹,

Chill²,

Djida³

2021

Preprint

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We give a representation for regular forms associated with dominated C 0 -semigroups which, in turn, characterises domination of C 0 -semigroups associated with regular forms. In addition, we prove a relationship between the positivity of (dominated) C 0 -semigroups and the locality of the associated forms.A characterisation of domination. Throughout, we let (Ω, A, µ) be a topological measure space. By this we mean that Ω is a topological space, A is the Borel σ-algebra, and µ is a (positive) Borel measure on (Ω, A). Without loss of generality, we assume that µ has full support in the sense that there exists no nonempty open subset U ⊆ Ω which has zero measure. Otherwise, we replace Ω with Ω \ U, where U is the largest open subset which has measure zero. All vector spaces in this note are complex vector spaces.We now state our main result: Theorem 1.1. Let a : D(a) × D(a) → C and a : D( a) × D( a) → C be two sesquilinear, Hermitean, closed, accretive, and densely defined forms on L 2 µ (Ω) such that the associated self-adjoint C 0 -semigroups (denoted by T and T respectively) are real. Assume that the semigroup T is positive, and that the form a is A-regular Date: 1st December 2021. 2020 Mathematics Subject Classification. 47D03, 31C15.

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Domination of nonlinear semigroups generated by regular, local Dirichlet forms

Chill,

Claus

2024

Annali di Matematica

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In this article we study perturbations of local, nonlinear Dirichlet forms on arbitrary topological measure spaces. As a main result, we show that the semigroup generated by a local, regular, nonlinear Dirichlet form $${\mathcal {E}}$$ E dominates the semigroup generated by another local functional $${\mathcal {F}}$$ F if, and only if, $${\mathcal {F}}$$ F is a specific zero order perturbation of $${\mathcal {E}}$$ E . On the way, we prove a nonlinear version of the Riesz–Markov representation theorem, we define an abstract boundary of a topological measure space, and apply the notion of nonlinear capacity. The main result helps to classify the perturbations that lie between Neumann and Dirichlet boundary conditions.

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Energy Spaces, Dirichlet Forms and Capacities in a Nonlinear Setting

Cited by 4 publications

References 13 publications

The Normal Contraction Property for Non-Bilinear Dirichlet Forms

The Normal Contraction Property for Non-Bilinear Dirichlet Forms

Domination of semigroups generated by regular forms

Domination of nonlinear semigroups generated by regular, local Dirichlet forms

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