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 R d , where 1 < p < d. We assume that the distance between anchoring points is at least ε and the characteristic diameter of cavities is αε , where α = α ( ε ) tends to 0 with ε . We also assume that anchoring points are asymptotically uniformly distributed as ε ↓ 0, and their number is asymptotic to a positive constant times ε 1 − d . The solution u = u ε 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 ε > 0. We show that the problem possesses a critical window characterized by τ := lim ε ↓ 0 α/α c ∈ (0 , ∞ ), where α c = ε 1 /γ and γ = d − p p − 1 . We prove that outside the unit sphere, as ε ↓ 0, the solution converges to A ∗ U for some constant A ∗ , where U ( x ) = min { 1 , | x | − γ } is the radial p -harmonic function outside the unit ball. Here the constant A ∗ equals 0 if τ = 0, while A ∗ = 1 if τ = ∞ . In the critical window where τ is positive and finite, A ∗ ∈ (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 εA ∗ that approximates the solution u ε in L ∞ ( R d ) and satisfies (cid:107)∇ u ε − ∇ u εA ∗ (cid:107) L p ( R d ) → 0 as ε ↓ 0. 31C45, 35B27, 35B40, 35J20, 35J25.