Weyl's Laplacian eigenvalue asymptotics for the measurable Riemannian structure on the Sierpinski gasket

Datum

10.05.2012

Zeit

14:50 - 15:50

Sprecher

Naotaka Kajino

Zugehörigkeit

Universität Bielefeld

Serie

TUD Mathematik AG Analysis & Stochastik

Sprache

en

Hauptthema

Mathematik

Andere Themen

Mathematik

Host

Prof. Dr. R. Schilling

Beschreibung

Bitte geänderte Anfangszeit beachten (am 09.05.2012 eingetragen)! On the Sierpinski gasket, Kigami [Math. Ann. 340 (2008), 781--804] has introduced the notion of the measurable Riemannian structure, with which the ``gradient vector fields" of functions, the ``Riemannian volume measure" and the ``geodesic metric" are naturally associated. Kigami has also proved in the same paper the two-sided Gaussian bound for the corresponding heat kernel, and I have further shown several detailed heat kernel asymptotics, such as Varadhan's asymptotic relation, in a recent paper [Potential Anal. 36 (2012), 67--115]. In the talk, Weyl's Laplacian eigenvalue asymptotics is presented for this case. The correct scaling order for the asymptotics of the eigenvalues is given by the Hausdorff dimension d of the gasket with respect to the ``geodesic metric", and in the limit of the eigenvalue asymptotics we obtain a constant multiple of the d-dimensional Hausdorff measure. Moreover, we will also see that this Hausdorff measure is Ahlfors regular with respect to the ``geodesic metric" but that it is singular to the ``Riemannian volume measure".

Links

• Webseite der AG Analysis & Stochastik

Letztmalig verändert: 09.05.2012, 16:21:19