A basic homogenization problem for the $p$-Laplacian in ${\mathbb R}^d$ perforated along a sphere: $L^\infty$ estimates

Abstract

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 $1 0$. We show that the problem possesses a critical window characterized by $\tau:=\lim_{\varepsilon \downarrow 0}\alpha /\alpha_c \in (0,\infty)$, where $\alpha_c=\varepsilon^{1/\gamma}$ and $\gamma= \frac{d-p}{p-1}.$ We prove that outside the unit sphere, as $\varepsilon\downarrow 0$, the solution converges to $A_*U$ for some constant $A_*$, where $U(x)=\min\{1,|x|^{-\gamma}\}$ is the radial $p$-harmonic function outside the unit ball. Here the constant $A_*$ equals 0 if $\tau=0$, while $A_*=1$ if $\tau=\infty$. In the critical window where $\tau$ 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 $\varepsilon \downarrow 0$.