Affiliations: University of Sannio, Department of Engineering, Piazza Roma, 21, 82100 Benevento, Italy. E-mail: giuseppe.cardone@unisannio.it | Institute of Mechanical Engineering Problems, V.O., Bolshoi pr., 61, 199178, St. Petersburg, Russia. E-mails: serna@snark.ipme.ru, srgnazarov@yahoo.co.uk | Département de Mathematiques, BP 239, Université H. Poincaré, Nancy 1, 54506 Vandoeuvre-lès-Nancy Cédex, France and Systems Research Institute of the Polish Academy of Sciences. E-mail: Jan.Sokolowski@iecn.u-nancy.fr, Poland
Note: [] The research of S.A.N. is partially supported by the Grant RFFI-06-01-257.
Note: [] The research of J.S. is partially supported by the Project CPER Lorraine MISN: Analyse, optimisation et controle 3 in France, and the Grant N51402132/3135 Ministerstwo Nauki i Szkolnictwa Wyzszego: Optymalizacja z wykorzystaniem pochodnej topologicznej dla przeplywow w osrodkach scisliwych in Poland.
Abstract: The Neumann problem for the Poisson equation is considered in a domain Ωε⊂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 uε and the Dirichlet integral ‖∇xuε; L2(Ωε)‖2 are evaluated and it is shown that main asymptotic term of uε 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.
