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 \(1<p<d.\) We assume that the distance between anchoring points is at least \(\varepsilon \) and the characteristic diameter of cavities is \(\alpha \varepsilon \), where \(\alpha =\alpha (\varepsilon )\) tends to 0 with \(\varepsilon \). We also assume that anchoring points are asymptotically uniformly distributed as \(\varepsilon \downarrow 0\), and their number is asymptotic to a positive constant times \(\varepsilon ^{1-d}\). The solution \(u=u^\varepsilon \) 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 \(\varepsilon >0\). 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 \(\Vert \nabla u^\varepsilon -\nabla u_{A_*}^\varepsilon \Vert _{L^{p}(\mathbb {R}^d)} \rightarrow 0\) as \(\varepsilon \downarrow 0\).