arXiv subjects
Publications and source records attributed to Guangzeng Yi.
We generalise the recent two-ends Furstenberg inequality due to Wang and the second author from lines to a family of transversal curves, and give a much simplified proof. As an application, we present a result pertaining to the Fourier decay of fractal measures on convex curves.
We prove an extension of the Furstenberg set theorem to families of graphs satisfying a transversality condition. We apply the result to derive bounds on $L^{p}$-norms of Fourier transforms of fractal measures supported on plane curves.
For all $s\in[0,1]$ and $t\in(0,s]\cup [2-s,2)$, we find the supremum of numbers $ω\in(0,2)$ such that $\text{I}_ω(μ\astσ) \lesssim 1$, where $μ$ is any Borel measure on $B(1)$ with $\text{I}_t(μ)\leq 1$ and $σ$ is any $(s,1)$-Frostman measure on a $C^2$-graph with non-zero curvature. As an application, we use this to show the sharp $L^6$-decay of Fourier transform of $σ$ when $s\in [\frac{2}{3}, 1]$.
This is the first in a series of papers devoted to the blow up analysis for the quenching phenomena in a parabolic MEMS equation. In this paper, we first give an optimal Hölder estimate for solutions to this equation by using the blow up method and some Liouville theorems on stationary two-valued caloric functions, and then establish a convergence theory for sequences of uniformly Hölder continuous solutions. These results are also used to prove a stratification theorem on the rupture set $\{u=0\}$.
Let $P \subset \mathbb{R}^{2}$ be a Katz-Tao $(δ,s)$-set, and let $\mathcal{L}$ be a Katz-Tao $(δ,t)$-set of lines in $\mathbb{R}^{2}$. A recent result of Fu and Ren gives a sharp upper bound for the $δ$-covering number of the set of incidences $\mathcal{I}(P,\mathcal{L}) = \{(p,\ell) \in P \times \mathcal{L} : p \in \ell\}$. In fact, for $s,t \in (0,1]$, $$ |\mathcal{I}(P,\mathcal{L})|_δ \lesssim_ε δ^{-ε-f(s,t)}, \qquad ε> 0,$$ where $f(s,t) = (s^{2} + st + t^{2})/(s + t)$. For $s,t \in (0,1]$, we characterise the near-extremal configurations $P \times \mathcal{L}$ of this inequality: we show that if $|\mathcal{I}(P,\mathcal{L})|_δ \approx δ^{-f(s,t)}$, then $P \times \mathcal{L}$ contains "cliques" $P' \times \mathcal{L}'$ satisfying $|\mathcal{I}(P',\mathcal{L}')|_δ \approx |P'|_δ|\mathcal{L}'|_δ$, $$|P'|_δ \approx δ^{-s^{2}/(s + t)} \quad \text{and} \quad |\mathcal{L}'|_δ \approx δ^{-t^{2}/(s + t)}.$$
In this paper, we study two PDEs that generalize the urban crime model proposed by Short \emph{et al}. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249-1267]. Our modifications are made under assumption of the spatial heterogeneity of both the near-repeat victimization effect and the dispersal strategy of criminal agents. We investigate pattern formations in the reaction-advection-diffusion systems with nonlinear diffusion over multi-dimensional bounded domains subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as the intrinsic near-repeat victimization rate $ε$ decreases and spatially nonconstant solutions emerge through bifurcation. Moreover, we find the wavemode selection mechanism through rigorous stability analysis of these nontrivial patterns, which shows that the only stable pattern must have wavenumber that maximizes the bifurcation value. Based on this wavemode selection mechanism, we will be able to precisely predict the formation of stable aggregates of the house attractiveness and criminal population density, at least when the diffusion rate $ε$ is around the principal bifurcation value. Our theoretical results also suggest that large domains support more stable aggregates than small domains. Finally, we perform extensive numerical simulations over 1D intervals and 2D squares to illustrate and verify our theoretical findings. Our numerics also include some interesting phenomena such as the merging of two interior spikes and the emerging of new spikes, etc. These nontrivial solutions can model the well observed aggregation phenomenon in urban criminal activities.