Related papers: On singularity of energy measures for symmetric di…
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…
In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their…
The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional in the class of $\mathbb{S}^2$-valued maps defined in cylindrical surfaces. The model naturally arises as a curved thin-film limit in the theories of…
We construct self-similar $p$-energy forms as normalized limits of discretized $p$-energies on a rich class of Laakso-type fractal spaces. Collectively, we refer to them as IGS-fractals, where IGS stands for (edge-)iterated graph systems.…
We construct a family of measures on $\bbR$ that are purely singular with respect to Lebesgue measure, and yet exhibit universal sine-kernel asymptotics in the bulk. The measures are best described via their Jacobi recursion coefficients:…
We consider a static, axially symmetric, and asymptotically flat exact solution of the Einstein vacuum equations, known as the gamma metric. This is characterized by two constant parameters $m$ and $\gamma$. We find that the total energy…
For $d\ge 2$ and $0<\beta<\alpha<2$, consider a family of non-local operators $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$ on $\mathbb{R}^d$, where $$ \mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{ \{z\in…
We analyze the Ericksen-Leslie system equipped with the Oseen-Frank energy in three space dimensions. The new concept of dissipative solutions is introduced. Recently, the author introduced the concept of measure-valued solutions to the…
In this paper, we derive explicit sharp two-sided estimates of the Dirichlet heat kernels for a class of symmetric subordinate diffusion processes with diffusive components in $C^{1, \alpha}(\alpha\in (0, 1])$ open sets in $\mathbb R^d$…
We construct symmetric self-similar diffusions with sub-Gaussian heat kernel estimates on two types of polygon carpets, which are natural generalizations of planner Sierpinski carpets (SC). The first ones are called perfect polygon carpets…
We construct good $p$-energy forms on metric measure spaces as pointwise subsequential limits of Besov-type $p$-energy functionals under certain geometric/analytic conditions. Such forms are often called Korevaar-Schoen $p$-energy forms in…
Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure:…
We prove a ${\Gamma}$-convergence result for the $p$-Dirichlet energy functional defined on maps from a smooth bounded domain $\Omega \subseteq \mathbb{R}^{n+k}$ to $\mathscr{N}$, a $(k-2)$-connected and smooth closed Riemannian manifold…
In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an…
On the unit interval (I), and the Sierpinski Gasket ($\mathcal{SG}$), the spectral decimation function of the Laplacian has similar properties that result in positive minimum spacing of eigenvalues. Other fractals, for example the level-3…
We construct a Sierpinski carpet like fractal, on which a self-similar diffusion with sub-Gaussian heat kernel estimate does not exist, in contrast to previous researches on the existence of such diffusions, on the generalized Sierpinski…
We provide general lower and upper bounds for Laplace Dirichlet heat kernel of convex $\mathcal C^{1,1}$ domains. The obtained estimates precisely describe the exponential behaviour of the kernels, which has been known only in a few special…
The mean square fluctuation and the expectation value of the stress-energy-momentum tensor of a neutral massive scalar field at finite temperature are determined near an infinite plane Dirichlet wall, and also near an infinite plane Neumann…
Let $\mathbb{A}$ and $\mathbb{A_{*}}$ be two non-degenerate spherical annuli in $\mathbb{R}^{n}$ equipped with the Euclidean metric and the weighted metric $|y|^{1-n}$, respectively. Let $\mathcal{F}(\mathbb{A},\mathbb{A_{*}})$ denote the…
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…