Paradoxes of Limit Passage in Solutions of Boundary Value Problems When Smooth Domains Are Approximated by Polygons
Vladimir Gilelevich Maz'ya, Serguei Nazarov, Boris A. Plamenevskij · Birkhäuser Basel eBooks · 2000
In the present chapter we approximate smooth domains by polygonal ones and analyze passing to the limit in solutions of boundary value problems. For this purpose we use an asymptotic approach. Though the techniques have a broader field of applications, we restrict ourselves to studying some particular boundary value problems in the theory of thin plates. More precisely, we consider a number of questions related to the known paradox that goes back to Sapondzhyan [2] and Babuška [2]. The case in point is the following example of instability: if one makes approximations of a thin circular plate by regular polygons with freely supported boundaries, one obtains the limiting solution that does not satisfy the freely supporting condition. We explain the asymptotic genesis of the Sapondzhyan-Babuska paradox, point a way to eliminate it and reveal some new phenomena of instability. 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.