arXiv subjects

Weicong Su

Publications and source records attributed to Weicong Su.

4 recordsLinked to original sources

arXiv2026

Geometric properties of Euclidean domains supporting trace inequalities

We investigate the geometric behavior of $\tau(E)$ for bounded finite-perimeter sets $E \subset \mathbb R^n$, where $\tau(E)$ is the trace constant introduced by Figalli--Maggi--Pratelli [Invent. Math. 2010]. This quantity is a key ingredient in proving a quantitative isoperimetric inequality with the optimal exponent. We first show that for every $\epsilon>0$ one can find a bounded open set $\Omega \subset \mathbb R^n$ that is very close to the unit ball $\mathbb B^n$ in the sense that $$ \tau(\mathbb B^n)>\tau(\Omega)>\tau(\mathbb B^n)-\epsilon \quad \text{and} \quad P(\Omega \Delta \mathbb B^n)\le C(n)\epsilon, $$ while at the same time the complement of $\Omega$ has infinitely many connected components. Thus, $\tau(\Omega)$ can be made arbitrarily close to $\tau(\mathbb B^n)$ even when $\Omega$ has highly intricate geometry. We then establish, under a mild additional hypothesis, the equivalence between a condition formulated in terms of $\tau$ and two classical criteria from the literature for open sets that admit trace inequalities. As a consequence, we obtain the John-type characterization of domains that support a trace inequality, assuming the ball separation property.

math.FA↗

arXiv2024

Sobolev trace inequalities on John domains and its applications

We prove that a trace inequality holds for John domains $\Omega$ satisfying $$ \mathcal H^{n-1}(\partial \Omega\setminus \partial_*\Omega)=0,$$ where $\partial_*\Omega$ denotes the measure-theoretic boundary, together with an upper density bound on $\partial \Omega$. This class of domains includes $(\epsilon,\,r)$-perimeter minimizers of Wulff perimeter $P_K$ which are close to the associated convex body $K$. Particularly, this result is established without requiring $\partial \Omega$ to be Ahlfors regular. As a consequence, we give an alternative proof for a crucial step in the quantitative Wulff inequality, thereby providing a meaningful commentary on the seminal work of Figalli, Maggi, and Pratelli \cite{FMP2010}.

math.OC↗

arXiv2024

Carrot John domains in variational problems

In this paper, we explore carrot John domains within variational problems, dividing our examination into two distinct sections. The initial part is dedicated to establishing the lower semicontinuity of the (optimal) John constant concerning Hausdorff convergence for bounded John domains. This result holds promising implications for both shape optimization problems and Techm\"uller theory. In the subsequent section, we demonstrate that an unbounded open set satisfying the carrot John condition with a center at $\infty$, appearing in the Mumford-Shah problem, can be covered by a uniformly finite number of unbounded John domains (defined conventionally through cigars). These domains, in particular, support Sobolev-Poincar\'e inequalities.

math.OC↗

arXiv2018

On the Peaks of a Stochastic Heat Equation on a Sphere with a Large Radius

For every $R>0$, consider the stochastic heat equation $\partial_{t} u_{R}(t\,,x)=\tfrac12 \Delta_{S_{R}^{2}}u_{R}(t\,,x)+\sigma(u_{R}(t\,,x)) \xi_{R}(t\,,x)$ on $S_{R}^{2}$, where $\xi_{R}=\dot{W_{R}}$ are centered Gaussian noises with the covariance structure given by $E [\dot{W_{R}}(t,x)\dot{W_{R}}(s,y)]=h_{R}(x,y)\delta_{0}(t-s)$, where $h_{R}$ is symmetric and semi-positive definite and there exist some fixed constants $-2< C_{h_{up}}< 2$ and $\frac{1}{2}C_{h_{up}}-1 0$ and $x\,,y \in S_{R}^{2}$, $(\log R)^{C_{h_{lo}}/2}=h_{lo}(R)\leq h_{R}(x,y) \leq h_{up}(R)=(\log R)^{C_{h_{up}}/2}$, $\Delta_{S_{R}^{2}}$ denotes the Laplace-Beltrami operator defined on $S_{R}^{2}$ and $\sigma:R \mapsto R$ is Lipschitz continuous, positive and uniformly bounded away from $0$ and $\infty$. Under the assumption that $u_{R,0}(x)=u_{R}(0\,,x)$ is a nonrandom continuous function on $x \in S_{R}^{2}$ and the initial condition that there exists a finite positive $U$ such that $\sup_{R>0}\sup_{x \in S_{R}^{2}}\vert u_{R,0}(x)\vert \le U$, we prove that for every finite positive $t$, there exist finite positive constants $C_{low}(t)$ and $C_{up}(t)$ which only depend on $t$ such that as $R \to \infty$, $\sup_{x \in S_{R}^{2}}\vert u_{R}(t\,,x)\vert$ is asymptotically bounded below by $C_{low}(t)(\log R)^{1/4+C_{h_{lo}}/4-C_{h_{up}}/8}$ and asymptotically bounded above by $C_{up}(t)(\log R)^{1/2+C_{h_{up}}/4}$ with high probability.

math.PR↗