Recent zbMATH articles in MSC 35P20https://zbmath.org/atom/cc/35P202021-06-15T18:09:00+00:00WerkzeugAsymptotic behaviour of the Steklov spectrum on dumbbell domains.https://zbmath.org/1460.352442021-06-15T18:09:00+00:00"Bucur, Dorin"https://zbmath.org/authors/?q=ai:bucur.dorin"Henrot, Antoine"https://zbmath.org/authors/?q=ai:henrot.antoine"Michetti, Marco"https://zbmath.org/authors/?q=ai:michetti.marcoSummary: We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.Asymptotics of eigenfunctions of the bouncing ball type of the operator \(\nabla D(x)\nabla\) in a domain bounded by semirigid walls.https://zbmath.org/1460.810322021-06-15T18:09:00+00:00"Klevin, A. I."https://zbmath.org/authors/?q=ai:klevin.a-iSummary: We consider the problem on the semiclassical spectrum of the operator \(\nabla D(x)\nabla\) with Bessel-type degeneration on the boundary of a two-dimensional domain (semirigid walls). It is well known that the asymptotic eigenfunctions associated with Lagrangian manifolds can be constructed using a modification of the Maslov canonical operator. We obtain asymptotic eigenfunctions associated with the simplest periodic trajectories of the corresponding Hamiltonian system with reflections on the domain boundary.