A Basic Homogenization Problem for the p-Laplacian in \(\mathbb {R}^d\) Perforated along a Sphere: \(L^\infty \) Estimates

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 \(\mathbb {R}^d\) ${\mathbb{R}}^d$ , where \(1<p<d.\) $1<p<d.$ We assume that the distance between anchoring points is at least \(\varepsilon \) $\epsilon$ and the characteristic diameter of cavities is \(\alpha \varepsilon \) $\alpha \epsilon$ , where \(\alpha =\alpha (\varepsilon )\) $\alpha =\alpha (\epsilon )$ tends to 0 with \(\varepsilon \) $\epsilon$ . We also assume that anchoring points are asymptotically uniformly distributed as \(\varepsilon \downarrow 0\) $\epsilon \downarrow 0$ , and their number is asymptotic to a positive constant times \(\varepsilon ^{1-d}\) ${\epsilon }^{1-d}$ . The solution \(u=u^\varepsilon \) $u=u^{\epsilon }$ 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 \(\varepsilon >0\) $\epsilon >0$ . We show that the problem possesses a critical window characterized by \(\tau :=\lim _{\varepsilon \downarrow 0}\alpha /\alpha _c \in (0,\infty )\) $\tau :={lim}_{\epsilon \downarrow 0}\alpha /{\alpha }_c\in (0,\infty )$ , where \(\alpha _c=\varepsilon ^{1/\gamma }\) ${\alpha }_c={\epsilon }^{1/\gamma }$ and \(\gamma = \frac{d-p}{p-1}.\) $\gamma =\frac{d-p}{p-1}.$ We prove that outside the unit sphere, as \(\varepsilon \downarrow 0\) $\epsilon \downarrow 0$ , the solution converges to \(A_*U\) $A_{\ast }U$ for some constant \(A_*\) $A_{\ast }$ , where \(U(x)=\min \{1,|x|^{-\gamma }\}\) $U(x)=min\{1,|x{|}^{-\gamma }\}$ is the radial p-harmonic function outside the unit ball. Here the constant \(A_*\) $A_{\ast }$ equals 0 if \(\tau =0\) $\tau =0$ , while \(A_*=1\) $A_{\ast }=1$ if \(\tau =\infty \) $\tau =\infty$ . In the critical window where \(\tau \) $\tau$ is positive and finite, \( A_*\in (0,1)\) $A_{\ast }\in (0,1)$ 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 \(u_{A_*}^\varepsilon \) $u_{A_{\ast }}^{\epsilon }$ that approximates the solution \(u^\varepsilon \) $u^{\epsilon }$ in \(L^{\infty }(\mathbb {R}^d)\) $L^{\infty }({\mathbb{R}}^d)$ and satisfies \(\Vert \nabla u^\varepsilon -\nabla u_{A_*}^\varepsilon \Vert _{L^{p}(\mathbb {R}^d)} \rightarrow 0\) $\Vert \mathrm{\nabla }{u}^{\epsilon }-\mathrm{\nabla }{u}_{A_{\ast }}^{\epsilon }{\Vert }_{L^p({\mathbb{R}}^d)}\to 0$ as \(\varepsilon \downarrow 0\) $\epsilon \downarrow 0$ .