English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

This thesis is about local and non-local Dirichlet forms on the Sierpi\'nskigasket and the Sierpi\'nski carpet. We are concerned with the following threeproblems in analysis on the Sierpi\'nski gasket and the Sierpi\'nski carpet.  First, a unified purely \emph{analytic} construction of local regularDirichlet forms on the Sierpi\'n-ski gasket and the Sierpi\'nski carpet. Wegive a purely analytic construction of a self-similar local regular Dirichletform on the Sierpi\'nski carpet using $\Gamma$-convergence of stable-likenon-local closed forms which gives an answer to an open problem in analysis onfractals. We also apply this construction on the Sierpi\'nski gasket.  Second, determination of walk dimension \emph{without} using diffusion.Although the walk dimension is a parameter that determines the behaviour ofdiffusion, we give two approaches to the determination of the walk dimension\emph{prior} to the construction of diffusion.  Third, approximation of local Dirichlet forms by non-local Dirichlet forms.We prove that non-local Dirichlet forms can approximate local Dirichlet formsas direct consequences of our construction of local Dirichlet forms. We alsoprove that on the Sierpi\'nski gasket the local Dirichlet form can be obtainedas a Mosco limit of non-local Dirichlet forms. Let us emphasize that we do\emph{not} need subordination technique based on heat kernel estimates.

Local and Non-Local Dirichlet Forms on the Sierpi\'nski Gasket and the Sierpi\'nski Carpet