Analytical properties of measure geometric Krein‐Feller‐operators on the real line

Abstract

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative d dµ d dν w.r.t. two atomless finite Borel measures µ and ν with compact supports supp µ ⊂ supp ν on the real line. This class of operators includes a generalization of the well-known Sturm-Liouville operator d dµ d dx as well as of the measure geometric Laplacian given by d dµ d dµ .We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gauß-Green-formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.

Metric‐based upscaling

“…Equations such as (1.65) have been studied in dimension 1 in [39,40] in order to introduce a measure-theoretic way of defining differential operators on fractal sets on the real line.…”

Dirichlet forms for singular diffusion in higher dimensions

“…Such singular diffusions in one dimension and the form approach were described in [8,9,[17][18][19], see also, e.g., [16] for form methods. As it turns out in one dimension, functions in the domain of the form (and hence also the operator) have to be affine on the complement of spt μ.…”

About Maxwell’s equations on fractal subsets of ℝ3

“…(see [30,34,35] and several references therein). Using measure theory people have introduced calculus on fractals [32,33]. It consists of defining a derivative as an inverse of an integral with respect to a measure and defining other operators using this derivative.…”