AI 中文总结
研究沿单位球面穿孔的d维球中p-拉普拉斯算子临界情形\(p = d\)的边值问题，通过参数\(\tau\)确定临界尺度，构造显式近似解，确定远离单位球面时解的收敛形式及不同\(\tau\)值下的常数\(A_*\)。
AI 中文摘要
我们研究穿孔区域\(B(0,\rho)\setminus\Gamma\subset \mathbb{R}^d\)中p-拉普拉斯算子的边值问题，其中\(\rho>1\)且\(\Gamma\)是位于单位球面附近的许多小紧致空洞的并集。空洞在\(\varepsilon\)尺度上分离，在球面上渐近等分布，基数为\(\varepsilon^{1 - d}\)阶。空洞直径为\(\alpha(\varepsilon)\varepsilon\)阶，其中\(\alpha(\varepsilon)\to0\)，其相对p-容量与相同直径球的相对p-容量可比。解在所有空洞上等于\(1\)，在\(\partial B(0,\rho)\)上等于\(0\)。我们关注临界情形\(p = d>1\)。通过参数\(\tau=\lim_{\varepsilon\downarrow0}[\varepsilon\log(1/\alpha(\varepsilon))]^{-1}\in[0,\infty]\)确定临界尺度。远离单位球面时，解收敛到\(A_*U_\rho\)，其中\(U_\rho(x)=\min\{1,1-\log |x|/\log\rho\}\)是\(B(0,\rho)\)中单位球的径向d-调和势。当\(\tau = 0\)时\(A_* = 0\)，当\(\tau=\infty\)时\(A_* = 1\)，\(0<\tau<\infty\)时明确。我们构造了一个显式近似解，在\(L^{\infty}\)和d-容量方面对足够小的\(\varepsilon\)近似原解。
英文摘要
We study a boundary value problem for the $p$-Laplacian in the perforated domain $B(0,ρ)\setminusΓ\subset \mathbb{R}^d$, where $ρ>1$ and $Γ$ is the union of many small compact cavities placed near the unit sphere. The cavities are separated at scale $\varepsilon$, asymptotically equidistributed on the sphere, and have cardinality of order $\varepsilon^{1-d}$. The cavities have diameters of order $α(\varepsilon)\varepsilon$, where $α(\varepsilon)\to0$, and their relative $p$-capacity is comparable to the relative p-capacity of a ball of the same diameter. The solution is required to equal $1$ on all cavities and $0$ on $\partial B(0,ρ)$. We focus on the critical case $p=d>1$. We identify the critical scale through the parameter $τ=\lim_{\varepsilon\downarrow0}[\varepsilon\log(1/α(\varepsilon))]^{-1}\in[0,\infty]$. Thus, $α(\varepsilon)=\exp[-(1+o(1))/(τ\varepsilon)]$ when $0<τ<\infty$. Away from the unit sphere, the solutions converge to $A_*U_ρ$, where $U_ρ(x)=\min\{1,1-\log |x|/\logρ\}$ is the radial $d$-harmonic potential of the unit ball in $B(0,ρ)$. The constant $A_*$ equals $0$ when $τ=0$, equals $1$ when $τ=\infty$, and is explicit for $0<τ<\infty$. We construct an explicit ansatz that approximates the solution for sufficiently small $\varepsilon$ in both $L^{\infty}$ and in terms of $d$-capacity.
Comments24 pages, 3 figures. arXiv admin note: text overlap with arXiv:2205.07133
