Melissa Tacy (ANU)

L^p estimates for eigenfunctions on manifolds with boundary

Abstract: Measuring the L^p mass of an eigenfunction allows us to determine its concentration properties. On a manifold without boundary such estimates follow from short time properties of the wave or semiclassical Schrödinger propagators. However the presence of a boundary opens the possibility for multiple reflections even in short time. This will lead to greater concentration of the eigenfunction (displayed by higher L^p norms). It is known, for example, that the whispering gallery modes show this higher concentration. In this talk I will introduce a method of studying the boundary L^p problem semiclassically by considering an exact solution to the boundary problem and an approximate solution to the ambient Helmholtz equation.