Homogenization limit for the boundary value problem with the P-laplace operator and a nonlinear third boundary condition on the boundary of the holes in a perforated domain

Abstract

We study the asymptotic behavior, as e → 0, of the solution ue to the boundary value problem for the equation –Δpue ≡ –div(|∇ue| p –2 ∇ue) = f, where p ∈ [2, n), in an e�periodically perforated domain Ω e ⊂ n , n ≥ 3, with the nonlinear third boundary condition ue + e –γ σ(x, ue) = e –γ g(x) specified on the boundary of the holes, where ue ≡ |∇ue| p –2 (∇ue, ν) and ν is the outward unit normal vector on the boundary of the holes. It is assumed that the diameter of the holes is equal to C0e α , where C0 > 0, α = , and γ = α(p – 1). Under these conditions, a homogenized problem con� taining a new nonlinear term is constructed and a the� orem is proved stating that the solution of the original problem converges as e → 0 to the solution of the homogenized problem. For the first time, this effect was noted in [3], where a similar problem was considered for Poisson’s equa� tion with n = 3 and α ∈ (2, 3] and another method was used to study the behavior of solutions as e → 0. Note that the method proposed applies to nonlinear prob� lems and perforated domains with a more complex perforation geometry. Let Ω be a bounded domain in n , n ≥ 3, with a smooth boundary ∂Ω, and let Y = , . Denote by G 0 the ball of radius 1 centered at the origin. For δ >0 and e > 0, we define the sets δB = {x| δ –1 x ∈ B} and = {x ∈ Ω| ρ(x, ∂Ω) > 2e}. Define a e = C 0 e α , where α = and C0 is a positive number, and let ∂νp ∂νp n