Note: this talk will not take place in the usual room. Location TBA.

Speaker: Naoki Saito (UC Davis)

Title: Laplacian eigenfunctions that do not feel the boundary: Theory, Computation, and Applications

Abstract: I will discuss Laplacian eigenfunctions defined on a Euclidean
domain of general shape, which “do not feel the boundary.”
These Laplacian eigenfunctions satisfy the Helmholtz equation inside the domain,
and can be extended smoothly and harmonically outside of the domain.
Although these eigenfunctions do not satisfy the usual Dirichlet or Neumann
boundary conditions, they can be computed via the eigenanalysis of the
integral operator (with the potential kernel) commuting with the Laplace
operator. Compared to directly solving the Helmholtz equations on such
domains, the eigenanalysis of this integral operator has several advantages
including the numerical stability and amenability to modern fast numerical
algorithms (e.g., the Fast Multipole Method).
In this talk, I will discuss their properties, the relationship with the
Krein-von Neumann self-adjoint extension of unbounded symmetric operators, and
certain applications including image extrapolation and characterization of
biological shapes.