Mathematical Sciences: Faculty Publications

Laplace Operators Related to Self-Similar Measures on Rd

Document Type

Article

Publication Date

10-15-2006

Publication Title

Journal of Functional Analysis

DOI

10.1016/j.jfa.2006.07.005

ISSN

0022-1236

Abstract

Given a bounded open subset Ω of Rd (d⩾1) and a positive finite Borel measure μ supported on ¯Ω with μ(Ω)>0, we study a Laplace-type operator Δμ that extends the classical Laplacian. We show that the properties of this operator depend on the multifractal structure of the measure, especially on its lower L∞-dimension dim̲∞(μ). We give a sufficient condition for which the Sobolev space H1/0(Ω) is compactly embedded in L2(Ω,μ), which leads to the existence of an orthonormal basis of L2(Ω,μ) consisting of eigenfunctions of Δμ. We also give a sufficient condition under which the Green's operator associated with μ exists, and is the inverse of −Δμ. In both cases, the condition dim̲∞(μ)>d−2 plays a crucial rôle. By making use of the multifractal Lq-spectrum of the measure, we investigate the condition dim̲∞(μ)>d−2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition.

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