Energy forms

2017/03/15 by Marcel Schmidt, Schmidt, Marcel · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1703.04883

openalex publication_date 2017/03/15 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

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:L⁰(m) → [0,∞], 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 → \mathbb R the energy of a function f satisfies E(C ∘ f) ≤ 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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