Homogenization for the $p$-Laplacian in a $d$-dimensional ball perforated along the unit sphere: the critical case $p=d$

Abstract

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 $\epsilon$, asymptotically equidistributed on the sphere, and have cardinality of order ${\epsilon }^{1-d}$. The cavities have diameters of order $\alpha (\epsilon )\epsilon$, where $\alpha (\epsilon )\to 0$, 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 $\tau ={\lim }_{\epsilon \downarrow 0}[\epsilon \log (1/\alpha (\epsilon )){]}^{-1}\in [0,\infty ]$. Thus, $\alpha (\epsilon )=\exp [-(1+o(1))/(\tau \epsilon )]$ when $0<\tau <\infty$. Away from the unit sphere, the solutions converge to $A_{\ast }{U}_{\rho }$, 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 $A_{\ast }$ 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 $\epsilon$ in both $L^{\infty }$ and in terms of $d$-capacity.

24 pages, 3 figures. arXiv admin note: text overlap with arXiv:2205.07133