Asymptotics of Solutions to Classical Boundary Value Problems in a Domain with Thin Cavities
Vladimir Gilelevich Maz'ya, С. А. Назаров, Boris A. Plamenevskij · Birkhäuser Basel eBooks · 2000
In the present chapter we describe the asymptotics of solutions some of boundary value problems in two- or three-dimensional domains perturbed near a closed curve or a segment. In Section 12.1 and 12.2 we consider the Neumann and Dirichlet problems for the Laplace operator in the exterior of a thin closed tube. The first boundary value problem for the Lamé system in the same geometric situation is studied in Section 12.3. Then we return to the Laplace operator and discuss the Dirichlet problem in a domain perturbed near an interval; Section 12.4 is devoted to the two-dimensional case and Section 12.5 to the three-dimensional one. In Section 12.6 we derive the asymptotics of a solution to the nonlinear equation Δ u + u 2 - f in a domain with boundary smoothed near an edge. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.