论文标题
$ {\ mathbb r}^d $沿球形打孔的$ p $ -laplacian的基本均质化问题：$ l^\ infty $估计
A basic homogenization problem for the $p$-Laplacian in ${\mathbb R}^d$ perforated along a sphere: $L^\infty$ estimates
论文作者
论文摘要
我们考虑了$ p $ -laplacian的边界价值问题，在小腔的外部摆构，它们都具有相同的$ p $ - 容量，并固定在$ \ mathbb {r}^d $中的单位球体上$α\ varepsilon $，其中$α=α（\ varepsilon）$倾向于使用$ \ varepsilon $。我们还假设锚定点在$ \ varepsilon \ downarrow 0 $上渐近地分布均匀分布，它们的数字是渐近的恒定时间$ \ varepsilon^{1-d} $。解决方案$ u = u = u^\ varepsilon $在所有腔中必须为1，而在无穷大时衰减为0。我们的目标是描述针对小$ \ varepsilon> 0 $的解决方案的行为。 We show that the problem possesses a critical window characterized by $τ:=\lim_{\varepsilon \downarrow 0}α/α_c \in (0,\infty)$, where $α_c=\varepsilon^{1/γ}$ and $γ= \frac{d-p}{p-1}.$ We prove that outside the unit sphere, as $ \ varepsilon \ downarrow 0 $，解决方案将某些常数$ a _*$收敛到$ a_*u $，其中$ u（x）= \ min \ {1，| x | x | x |^{ - γ} \} $是radial $ p $ p $ - harm-harm-harm-harm-harmornocon untel单位球。如果$τ= 0 $，则常数$ a _*$等于0，而$ a _*= 1 $如果$τ= \ infty $。在关键窗口中，$τ$是正面和有限的，$ a _*\ in（0,1）$是根据问题的参数明确计算的。我们还评估了上述所有三个案例的限制$ p $容量。我们的关键新工具是构建明确的ansatz函数$ u_ {a _*}^\ varepsilon $，该$近似于解决方案$ l^{\ infty}（\ mathbb {r}^d）$ in $ l^{\ mathbb {r}^d）$ in u_ {a _*}^\ varepsilon \ | _ {l^{p}（\ Mathbb {r}^d）} \ to 0 $ as as $ \ varepsilon \ downarrow 0 $。
We consider a boundary value problem for the $p$-Laplacian, posed in the exterior of small cavities that all have the same $p$-capacity and are anchored to the unit sphere in $\mathbb{R}^d$, where $1<p<d.$ We assume that the distance between anchoring points is at least $\varepsilon$ and the characteristic diameter of cavities is $α\varepsilon$, where $α=α(\varepsilon)$ tends to 0 with $\varepsilon$. We also assume that anchoring points are asymptotically uniformly distributed as $\varepsilon \downarrow 0$, and their number is asymptotic to a positive constant times $\varepsilon^{1-d}$. The solution $u=u^\varepsilon$ is required to be 1 on all cavities and decay to 0 at infinity. Our goal is to describe the behavior of solutions for small $\varepsilon>0$. We show that the problem possesses a critical window characterized by $τ:=\lim_{\varepsilon \downarrow 0}α/α_c \in (0,\infty)$, where $α_c=\varepsilon^{1/γ}$ and $γ= \frac{d-p}{p-1}.$ We prove that outside the unit sphere, as $\varepsilon\downarrow 0$, the solution converges to $A_*U$ for some constant $A_*$, where $U(x)=\min\{1,|x|^{-γ}\}$ is the radial $p$-harmonic function outside the unit ball. Here the constant $A_*$ equals 0 if $τ=0$, while $A_*=1$ if $τ=\infty$. In the critical window where $τ$ is positive and finite, $ A_*\in(0,1)$ is explicitly computed in terms of the parameters of the problem. We also evaluate the limiting $p$-capacity in all three cases mentioned above. Our key new tool is the construction of an explicit ansatz function $u_{A_*}^\varepsilon$ that approximates the solution $u^\varepsilon$ in $L^{\infty}(\mathbb{R}^d)$ and satisfies $\|\nabla u^\varepsilon-\nabla u_{A_*}^\varepsilon \|_{L^{p}(\mathbb{R}^d)} \to 0$ as $\varepsilon \downarrow 0$.
