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Energy forms

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arxiv 1703.04883 v1 pith:KZHXY7NL submitted 2017-03-15 math.FA

Marcel Schmidt This is my paper · ORCID
classification math.FA
keywords energyformscontractionfunctioncontinuitymathbbmeasurableproperty
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abstract

In this thesis we study energy forms. These are quadratic forms on the space of real-valued measurable $m$-a.e. determined functions $$E:L^0(m) \to [0,\infty],$$ which assign to a measurable function $f$ its energy $E(f)$. Their two defining characteristics are a contraction property and some form of continuity. The contraction property demands that for each normal contraction $C:\mathbb R \to \mathbb R$ the energy of a function $f$ satisfies $$E(C \circ f) \leq E(f).$$ This is an abstract formulation of the postulate that cutting off fluctuations of a function (which is thought to describe some physical quantity) decreases its energy. The continuity assumption that we impose on energy forms is lower semicontinuity with respect to local convergence in measure. We develop the basic theory of energy forms and then investigate their extensions and global properties.

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Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Nonlinear resistance forms

    math.FA 2025-07 conditional novelty 7.0 of 10

    A new abstract class of nonlinear resistance forms is defined, with resistance metrics that satisfy the triangle inequality and add over serial circuits, unifying Kigami and p-resistance forms.

  2. The extended Dirichlet space and criticality theory for nonlinear Dirichlet forms

    math.FA 2025-01 accept novelty 7.0 of 10

    Nonlinear Dirichlet forms have an extended Dirichlet space, and their criticality and subcriticality are characterized by Hardy and Poincaré inequalities and by the norm of the constant function 1.

  3. Characterizations of $p$-Parabolicity on Graphs

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    Several characterizations of p-parabolicity (Ahlfors, Kelvin-Nevanlinna-Royden, Khas'minskiĭ, Poincaré) hold on infinite locally summable graphs for p-energy functionals, with applications to the p-Laplacian obstacle ...

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