论文标题
内部聚合模型具有多种来源和障碍物问题
Internal aggregation models with multiple sources and obstacle problems on Sierpinski gaskets
论文作者
论文摘要
We consider the doubly infinite Sierpinski gasket graph $SG_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs $SG_n=2^{-n}SG_0$, for every $n\in \mathbb{N}$, we investigate the limit shape of three aggregation models with initial configuration $σ_n$ of particles supported on multiple顶点。所考虑的模型是：可划分的沙门，其中多余的质量分布在顶点之间，直到每个顶点稳定，并且质量较小，并且质量较小或等于一个，其中颗粒会随机行走直到找到空位点，而转子聚集粒子进行粒子进行确定性的随机步行，直到找到空位置为止。 We denote by $SG=cl(\cup_{n=0}^{\infty} SG_n)$ the infinite Sierpinski gasket, which is a closed subset of $\mathbb{R}^2$, for which $SG_n$ represents the level-n approximating graph, and we consider a continuous function $σ:SG\to\mathbb{N}$.对于$σ$，我们解决了障碍问题，我们将非复合设置$ d \ subset sg $描述为分形$ sg $上的自由边界问题的解决方案。 If the discrete particle configurations $σ_n$ on the approximating graphs $SG_n$ converge pointwise to the continuous function $σ$ on the limit set $SG$, we prove that, as $n\to\infty$, the scaling limits of the three aforementioned models on $SG_n$ starting with initial particle configuration $σ_n$ converge to the deterministic solution $D$ of the free boundary problem on the限制设置$ sg \ subset \ mathbb {r}^2 $。对于$ d $，我们还研究边界规律性属性。
We consider the doubly infinite Sierpinski gasket graph $SG_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs $SG_n=2^{-n}SG_0$, for every $n\in \mathbb{N}$, we investigate the limit shape of three aggregation models with initial configuration $σ_n$ of particles supported on multiple vertices. The models under consideration are: divisible sandpile in which the excess mass is distributed among the vertices until each vertex is stable and has mass less or equal to one, internal DLA in which particles do random walks until finding an empty site, and rotor aggregation in which particles perform deterministic counterparts of random walks until finding an empty site. We denote by $SG=cl(\cup_{n=0}^{\infty} SG_n)$ the infinite Sierpinski gasket, which is a closed subset of $\mathbb{R}^2$, for which $SG_n$ represents the level-n approximating graph, and we consider a continuous function $σ:SG\to\mathbb{N}$. For $σ$ we solve the obstacle problem and we describe the noncoincidence set $D\subset SG$ as the solution of a free boundary problem on the fractal $SG$. If the discrete particle configurations $σ_n$ on the approximating graphs $SG_n$ converge pointwise to the continuous function $σ$ on the limit set $SG$, we prove that, as $n\to\infty$, the scaling limits of the three aforementioned models on $SG_n$ starting with initial particle configuration $σ_n$ converge to the deterministic solution $D$ of the free boundary problem on the limit set $SG\subset\mathbb{R}^2$. For $D$ we also investigate boundary regularity properties.
