Homogenization for the $p$-Laplacian in a $d$-dimensional ball perforated along the unit sphere: the critical case $p=d$
We study a boundary value problem for the $p$-Laplacian in the perforated domain $B(0,\rho)\setminus\Gamma\subset \mathbb{R}^d$, where $\rho>1$ and $\Gamma$ 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 $\alpha(\varepsilon)\varepsilon$, where $\alpha(\varepsilon)\to0$, 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 $\partial B(0,\rho)$. We focus on the critical case $p=d>1$. We identify the critical scale through the parameter $\tau=\lim_{\varepsilon\downarrow0}[\varepsilon\log(1/\alpha(\varepsilon))]^{-1}\in[0,\infty]$. Thus, $\alpha(\varepsilon)=\exp[-(1+o(1))/(\tau\varepsilon)]$ when $0<\tau<\infty$. Away from the unit sphere, the solutions converge to $A_*U_\rho$, where $U_\rho(x)=\min\{1,1-\log |x|/\log\rho\}$ is the radial $d$-harmonic potential of the unit ball in $B(0,\rho)$. The constant $A_*$ equals $0$ when $\tau=0$, equals $1$ when $\tau=\infty$, and is explicit for $0<\tau<\infty$. We construct an explicit ansatz that approximates the solution for sufficiently small $\varepsilon$ in both $L^{\infty}$ and in terms of $d$-capacity.