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

2022/05/14 by Gordon, Peter V., Nazarov, Fedor, Peres, Yuval
#35J66 #35J92 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2205.07133

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 ℝ^d, where 10. We show that the problem possesses a critical window characterized by τ:=lim_{ε \downarrow 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\downarrow 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 \downarrow 0.

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