The obstacle scattering for the biharmonic equation

Abstract

In this paper, we consider the obstacle scattering problem for biharmonic equations with a Dirichlet boundary condition in both two and three dimensions. Some basic properties are first derived for the biharmonic scattering solutions, which leads to a simple criterion for the uniqueness of the direct problem. Then a new type far-field pattern is introduced, where the correspondence between the far-field pattern and scattered field is established. Based on these properties, we prove the well-posedness of the direct problem in associated function spaces by utilizing the boundary integral equation method, which relys on a natural decomposition of the biharmonic operator and the theory of the pseudodifferential operator. Furthermore, the inverse problem for determining the obstacle is studied. By establishing some novel reciprocity relations between the far-field pattern and scattered field, we show that the obstacle can be uniquely recovered from the measurements at a fixed frequency.