Variational Characterization of Fractal Neumann Eigenvalues
Primary Investigator (PI) Name
Eric Stachura
Department
CSM - Mathematics
Abstract
A fractal is a self-similar, infinitely repeating geometric object. These exhibit many counterintuitive mathematical properties. One such property is a fractal dimension, where the dimension of an object does not necessarily need to be an integer. A method called dimensional regularization can be used to transform fractional integrals into integrals in the more intuitive Euclidean space. We have shown that in this fractal space, many properties of solutions to partial differential equations continue to hold. In particular, we have shown that in the Neumann problem for a fractal version of the Laplace equation, there is a variational characterization of the Neumann eigenvalues via a corresponding notion of the Rayleigh quotient. This enables easier computation of eigenvalues, either by numerical methods or by hand.
Variational Characterization of Fractal Neumann Eigenvalues
A fractal is a self-similar, infinitely repeating geometric object. These exhibit many counterintuitive mathematical properties. One such property is a fractal dimension, where the dimension of an object does not necessarily need to be an integer. A method called dimensional regularization can be used to transform fractional integrals into integrals in the more intuitive Euclidean space. We have shown that in this fractal space, many properties of solutions to partial differential equations continue to hold. In particular, we have shown that in the Neumann problem for a fractal version of the Laplace equation, there is a variational characterization of the Neumann eigenvalues via a corresponding notion of the Rayleigh quotient. This enables easier computation of eigenvalues, either by numerical methods or by hand.