Spectral zeta functions of fractals and the complex dynamics of polynomials

We obtain formulas for the spectral zeta function of the Laplacian on symmetric finitely ramified fractals, such as the Sierpinski gasket, and a fractal Laplacian on the interval. These formulas contain a new type of zeta function associated with a polynomial (rational functions also can appear in this context). It is proved that this zeta function has a meromorphic continuation to a half plain with poles contained in an arithmetic progression. It is shown as an example that the Riemann zeta function is the zeta functions of a quadratic polynomial, which is associated with the Laplacian on an interval. The spectral zeta function of the Sierpinski gasket is a product of the zeta function of a polynomial and a geometric part; the poles of the former are canceled by the zeros of the latter. A similar product structure was discovered by M.L. Lapidus for self-similar fractal strings

Energy measure closability for Dirichlet forms

We consider symmetric Dirichlet forms on locally compact and non-locally compact spaces and provide an elementary proof for their closability with respect to energy dominant measures. We also discuss how to use known potential theoretic results to furnish an alternative proof of this theorem

Vector analysis for Dirichlet forms and quasilinear PDE and SPDE on metric measure spaces

Starting with a regular symmetric Dirichlet form on a locally compact separable metric space $X$, our paper studies elements of vector analysis, $L_p$-spaces of vector fields and related Sobolev spaces. These tools are then employed to obtain existence and uniqueness results for some quasilinear elliptic PDE and SPDE in variational form on $X$ by standard methods. For many of our results locality is not assumed, but most interesting applications involve local regular Dirichlet forms on fractal spaces such as nested fractals and Sierpinski carpets