Note on the Dirichlet Approximation Theorem

由冠定理得到的Besov-Dirichlet空间中的循环性

Cyclicity in Besov-Dirichlet spaces from the Corona Theorem

该研究借助Tolokonnikov冠定理，推导得出Besov-Dirichlet空间里循环性相关的两项结论。

A trace theorem for Dirichlet forms on fractals

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 their boundaries, where boundaries mean the triangles and rectangles which confine gaskets and carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields, which behave as the appropriate fractal diffusion within each fractal component of the field.

A partial reciprocal of Dirichlet Lagrange Theorem detected by Jets

We study the stability of an equilibrium point in a conservative Hamiltonian system in the case that equilibrium is not a minimum of the potential energy and this fact is shown by a jet of this function. Thanks to a modification of a result of Krasovskii, we prove that for a large class of systems under these conditions equilibrium is unstable and there is an asymptotic trajectory to that point.

Liouville theorem on a half-space for biharmonic problem with Dirichlet boundary condition

We investigate here the nonlinear elliptic Hénon type equation: $$\D^{2} u= |x|^a|u|^{p-1}u \; \,\,\mbox{in}\,\,\,\, \R^{n}_{+}, \quad \quad u =\frac{\partial u}{\partial x_n} = 0 \quad \mbox{in}\,\,\,\, \partial \R^{n}_{+},$$ with $p>1$ and $n\geq 2$. In particular, we prove some Liouville type theorems for stable at infinity solutions. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.

Havin-Mazya type uniqueness theorem for Dirichlet spaces

Let $μ$ be a positive finite Borel measure on the unit circle. The associated Dirichlet space $\mathcal{D}(μ)$ consists of holomorphic functions on the unit disc whose derivatives are square integrable when weighted against the Poisson integral of $μ$. We give a sufficient condition on a Borel subset $E$ of the unit circle which ensures that $E$ is a uniqueness set for $\mathcal{D}(μ)$. {We also give somes examples of positive Borel measures $μ$ and uniqueness sets for $\mathcal{D}(μ)$.}

Generalized Cesàro operators on Dirichlet-type spaces

In this note, we introduce and study a new kind of generalized Cesàro operators $\mathcal{C}_μ$, induced by a positive Borel measure $μ$ on $[0, 1)$, between the Dirichlet-type spaces. We characterize the measures $μ$ for which $\mathcal{C}_μ$ is bounded (compact) from one Dirichlet-type space $\mathcal{D}_α$ into another one $\mathcal{D}_β$.