Quasimodes for the Laplace operator and glancing hypersurfaces. (English) Zbl 0794.35030

Popov, Georgi S.

Microlocal analysis and nonlinear waves, Proc. Workshop, IMA Program Nonlinear Waves, Minneapolis/MN (USA) 1988-89, IMA Vol. Math. Appl. 30, 167-178 (1991).

Summary: [For the entire collection see Zbl 0758.00007.]
This paper is concerned with the construction of a quasimode for the Laplace operator in a bounded domain \(\Omega\) in \(\mathbb{R}^ n\), \(n\geq 2\), which a Dirichlet (Neumann) boundary condition. The quasimode is associated either with a closed gliding ray on the boundary or with a closed broken ray in \(T^*\Omega\). The frequency set of the quasimode consists of the conic hull of the union of the bicharacteristics of the cosphere bundle \(S^*\Omega\) issuing from a family of invariant tori of the billiard ball map. To construct a quasimode near a gliding ray we find a global symplectic normal form for a pair of glancing hypersurfaces.

MSC:

35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35A27	Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
35P05	General topics in linear spectral theory for PDEs

Keywords:

construction of a quasimode; Laplace operator in a bounded domain; frequency set