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.