2026/07/20 by Peter V. Gordon, Yuval Peres
#math.AP
We study a boundary value problem for the p-Laplacian in the perforated domain B(0,ρ)∖Γ⊂ ℝd, where ρ>1 and Γ is the union of many small compact cavities placed near the unit sphere. The cavities are separated at scale ε, asymptotically equidistributed on the sphere, and have cardinality of order ε1-d. The cavities have diameters of order α(ε)ε, where α(ε)→0, and their relative p-capacity is comparable to the relative p-capacity of a ball of the same diameter. The solution is required to equal 1 on all cavities and 0 on ∂ B(0,ρ). We focus on the critical case p=d>1. We identify the critical scale through the parameter τ=limε\downarrow0[εlog(1/α(ε))]-1∈[0,∞]. Thus, α(ε)=exp[-(1+o(1))/(τε)] when 0<τ<∞. Away from the unit sphere, the solutions converge to A_*Uρ, where Uρ(x)=min\1,1-log |x|/logρ\ is the radial d-harmonic potential of the unit ball in B(0,ρ). The constant A_* equals 0 when τ=0, equals 1 when τ=∞, and is explicit for 0<τ<∞. We construct an explicit ansatz that approximates the solution for sufficiently small ε in both L∞ and in terms of d-capacity.