Spectral Asymptotics of Some One-Dimensional Fractal Laplacians
Abstract or Description
The spectral dimension of the Laplacian deﬁned by a measure has been shown to be closely related to heat kernel estimates, which under suitable conditions determine whether wave propagates with ﬁnite or inﬁnite speed. We observe that some self-similar measures deﬁned by ﬁnite or inﬁnite iterated function systems with overlaps satisfy certain “essentially ﬁnite type condition”, which allows us to extract useful measure-theoretic properties of iterates of the measure. We develop a technique to obtain, under this condition, a closed formula for the spectral dimension of the Laplacian. Earlier results for fractal measures with overlaps rely on Strichartz second-order identities, which are not satisﬁed by the measures we consider here. This is a joint work with Wei Tang and Yuanyuan Xie.
Southeastern-Atlantic Regional Conference on Differential Equations (SEARCDE)
"Spectral Asymptotics of Some One-Dimensional Fractal Laplacians."
Department of Mathematical Sciences Faculty Presentations.