Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary

Abstract

The Neumann problem for the Poisson equation is considered in a domain Ω_ε⊂Rⁿ with boundary components posed at a small distance ε>0 so that in the limit, as ε→0⁺, the components touch each other at the point 𝒪 with the tangency exponent 2m≥2. Asymptotics of the solution u_ε and the Dirichlet integral ‖∇_xu_ε; L²(Ω_ε)‖² are evaluated and it is shown that main asymptotic term of u_ε and the existence of the finite limit of the integral depend on the relation between the spatial dimension n and the exponent 2m. For example, in the case n<2m−1 the main asymptotic term becomes of the boundary layer type and the Dirichlet integral has no finite limit. Some generalizations are discussed and certain unsolved problems are formulated, in particular, non-integer exponents 2m and tangency of the boundary components along smooth curves.