A Basic Homogenization Problem for the p -Laplacian in ℝ^d Perforated along a Sphere: L^∞ 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 10 . We show that the problem possesses a critical window characterized by τ :=lim _ε↓ 0α /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_A_*^ε that approximates the solution u^ε in L^∞(ℝ^d) and satisfies ‖∇ u^ε -∇ u_A_*^ε‖ _L^p(ℝ^d)→ 0 as ε↓ 0 .