Analysis and Its Applications Seminar
Approximating Eigenvalues of Self Adjoint Dirichlet Problems Over Domains of Irregular Shape - Preliminary Report
Abstract:
Two-sided approximations to single eigenvalues of the Laplace operator over bounded Lipschitzian Graph domains are obtainable by the method of point solutions. Each eigenvalue must be estimated separately, and to tell which of the ordered eigenvalues is approximated, additional a priori information, such as the Weyl asymptotics, must be invoked. This method uses solutions of the homogeneous Helmholtz equation as trial functions, but does not require satisfaction of any boundary conditions.
The Heat Kernel Asymptotics in a Polyhedron
Abstract:
The contribution of a vertex to the small time asymptotics of the heat trace in a polyhedron is rather complicated. I will give an expression for this contribution in terms of a certain function that is related to Brownian motion. I will also discuss the history of the problem.
Nonlinear Convective Instability of Fronts in Reaction-Diffusion Systems
Abstract:
Fronts are traveling waves in spatially extended systems that connect two different spatially homogeneous rest states. If the rest state behind the front becomes unstable, then the front will also destabilize. On the linear level there exists an exponentially weighted norm that stabilizes the front; in other words, the instability of the front in the co-moving frame is convective since perturbations are pushed away from the interface of the front.