相关论文: Construction of Local Regular Dirichlet Form on th…
We give a purely analytic construction of a self-similar local regular Dirichlet form on the Sierpi\'nski carpet using approximation of stable-like non-local closed forms which gives an answer to an open problem in analysis on fractals.
This thesis is about local and non-local Dirichlet forms on the Sierpi\'nski gasket and the Sierpi\'nski carpet. We are concerned with the following three problems in analysis on the Sierpi\'nski gasket and the Sierpi\'nski carpet. First, a…
We construct equivalent semi-norms of non-local Dirichlet forms on the Sierpi\'nski gasket and apply these semi-norms to a convergence problem and a trace problem. We also construct explicitly a sequence of non-local Dirichlet forms with…
Let $\Gamma_n$ denote the $n$-th level Sierpi\'nski graph of the Sierpi\'nski gasket $K$. We consider, for any given conductance $(a_0, b_0, c_0)$ on $\Gamma_0$, the Dirchlet form ${\mathcal E}$ on $K$ obtained from a recursive construction…
We construct a strongly local regular Dirichlet form on the golden ratio Sierpinski gasket, which is a self-similar set without any finitely ramified cell structure, via a study on the trace of electrical networks on an infinite graph. The…
We study the convergence of resistance metrics and resistance forms on a converging sequence of spaces. As an application, we study the existence and uniqueness of self-similar Dirichlet forms on Sierpinski gaskets with added rotated…
We construct symmetric self-similar Dirichlet forms on unconstrained Sierpinski carpets, which are natural extension of planar Sierpinski carpets by allowing the small cells to live off the $1/k$ grids. The intersection of two cells can be…
The resolvent convergence of self-adjoint operators via the technique of $\Gamma$-convergence of quadratic forms is adapted to incorporate complex Hilbert spaces. As an application, we find effective operators to the Dirichlet Laplacian…
By using the analytic tools of Dirichlet forms, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along a fractal (which can be considered as a simplified rough porous…
We present a new approach to the theory of k-forms on self-similar fractals. We work out the details for two examples, the standard Sierpinski gasket and the 3-dimensional Sierpinski gasket, but the method is expected to be effective for…
We give a short, self-contained analytic proof of the existence of self-similar Dirichlet forms on pillow-type carpets, a family of infinitely ramified fractals that includes the Sierpi\'nski carpet.
We prove the existence of a strongly local, regular, self-similar Dirichlet form with a sub-Gaussian heat kernel estimate on an unconstrained Sierpinski carpet in $\mathbb{R}^3$. In the setting under consideration, the walk dimension $d_W$…
The {\it Sierpi\'nski fractal} or {\it Sierpi\'nski gasket} $\Sigma$ is a familiar object studied by specialists in dynamical systems and probability. In this paper, we consider a graph $S_n$ derived from the first $n$ iterations of the…
We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on the Sierpinski gaskets and the Sierpinski carpets to…
In this paper, we study discrete approximations of semi-Dirichlet forms obtained by adding non-symmetric drift terms, expressed in terms of mutual energy measures, to resistance forms whose associated resistance metric spaces are compact.…
Koskela and Zhou have proven that, on the harmonic Sierpinski gasket with Kusuoka's measure, the "natural" Dirichlet form coincides with Cheeger's energy. We give a different proof of this result, which uses the properties of the Lyapounov…
We consider criteria for the differentiability of functions with continuous Laplacian on the Sierpinski Gasket and its higher-dimensional variants $SG_N$, $N>3$, proving results that generalize those of Teplyaev. When $SG_N$ is equipped…
We provide a definition of integral, along paths in the Sierpinski gasket K, for differential smooth 1-forms associated to the standard Dirichlet form K. We show how this tool can be used to study the potential theory on K. In particular,…
J. Kigami has laid the foundations of what is now known as analysis on fractals, by allowing the construction of an operator of the same nature of the Laplacian, defined locally, on graphs having a fractal character. The Sierpinski gasket…
We prove that the Sierpi\'nski gasket is non-removable for quasiconformal maps, thus answering a question of Bishop. The proof involves a new technique of constructing an exceptional homeomorphism from $\mathbb R^2$ into some non-planar…