论文标题
多参数周期性问题问题的奇异行为
Singular behavior for a multi-parameter periodic Dirichlet problem
论文作者
论文摘要
我们认为在定期穿孔的域中，对于泊松方程的一个差异问题。域的几何形状由两个参数控制：一个实际数字$ε> 0 $与孔的半径和地图$ ϕ $成比例，该$ $ ϕ $建模了孔的形状。因此，如果$ g $表示dirichlet边界基准，而托儿所基准为$ f $，我们为每个四倍体$（ε，ϕ，g，f）$都有一个解决方案。我们的目的是研究解决方案如何取决于$（ε，ϕ，g，f）$，尤其是当$ε$很小并且孔窄到点时。与以前的工作相反，我们没有引入这样的假设，即$ f $在基本周期性单元上的积分为零。这带来了$ε$接近$ 0 $的某种单一行为。我们表明，当环境空间的尺寸$ n $大于或等于$ 3 $时，可以用四倍$（ε，ϕ，g，f）乘以因子$ 1/ε^{n-2} $乘以二倍$（ε，ϕ，g，f）的分析图来表示解决方案的适当限制。如果尺寸$ n = 2 $，我们必须添加$ \logε$ times $ f/2π$的积分。
We consider a Dirichlet problem for the Poisson equation in a periodically perforated domain. The geometry of the domain is controlled by two parameters: a real number $ε>0$ proportional to the radius of the holes and a map $ϕ$, which models the shape of the holes. So, if $g$ denotes the Dirichlet boundary datum and $f$ the Poisson datum, we have a solution for each quadruple $(ε,ϕ,g,f)$. Our aim is to study how the solution depends on $(ε,ϕ,g,f)$, especially when $ε$ is very small and the holes narrow to points. In contrast with previous works, we don't introduce the assumption that $f$ has zero integral on the fundamental periodicity cell. This brings in a certain singular behavior for $ε$ close to $0$. We show that, when the dimension $n$ of the ambient space is greater than or equal to $3$, a suitable restriction of the solution can be represented with an analytic map of the quadruple $(ε,ϕ,g,f)$ multiplied by the factor $1/ε^{n-2}$. In case of dimension $n=2$, we have to add $\log ε$ times the integral of $f/2π$.
