Local Geometry And Application Of Elastic Non-radiating Source And Transmission Eigenvalue Problem At Corner

Elastic wave scattering theory includes elastic obstacle scattering,elastic medium scattering and elastic source scattering,they have important applications in the fields of geophysics,geological exploration and earthquake early warning.Therefore,the study of elastic wave scattering has important scientific significance.In this thesis,we consider the elastic wave non-radiating source scattering problem and the elastic transmission eigenvalue problem.For the elastic wave non-radiating source scattering problem,if the support of the elastic source contains convex conic or polyhedron corners,and the elastic source has H?lder regularity at the corners,we use complex geometric optics solutions to perform fine asymptotic analysis in phase space to prove that the elastic source must vanish at the corners.The local geometric properties is of great significance,it establishes the relationship between the nonradiating source and the geometry of source support.It is also shown that when the source support contains convex conic or polyhedron corners,and the physics field of the elastic source does not disappear at the geometric singularity,non-radiating is impossible.In terms of practical applications,we take advantage of the local geometric property,under a single far-field measurement,the uniqueness of the source support shape inversion of the elastic wave inverse source problem is established by a contradiction.For the elastic transmission eigenvalue problem,if the area involved in this transmission eigenvalue problem contains convex conic or polyhedron corners,when the transmission eigenfunction has H?lder regularity or can be approximated to some degree by the Herglotz wave functions,we analyze the order of asymptotic parameters of complex geometric optics solutions in phase space,then it is proved that the corresponding transmission eigenfunction vanishes at the corners.Under generic conditions,by virtue of the local geometric property of the transmission eigenfunction and fine analysis with a method of contradiction,we establish the uniqueness of shape inversion for the inverse scattering problem of elastic medium by a single far-field measurement.