Functional inequalities for Dunkl operators with applications
File(s)
Author(s)
Velicu, Cornel Andrei
Type
Thesis
Abstract
In this thesis we thoroughly study several families of classical functional inequalities in the setting of Dunkl operators. The following families are considered: Sobolev, Hardy, Gagliardo-Nirenberg, logarithmic Sobolev, Poincare inequalities. In each case, several versions of the inequality are discussed, as well as related results. In the case of the Sobolev inequality for p=2 we provide bounds on the optimal constant, and in the process we also determine the precise optimal constants for the equivalent weighted Sobolev inequality for the usual gradient. In terms of spectral inequalities, we obtain Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for Schrodinger operators with Dunkl laplacian.
As an application of the Gagliardo-Nirenberg inequality, we prove an existence and uniqueness result for the Cauchy problem with small data of the nonlinear damped wave equation. Furthermore, we prove ergodicity for an infinite dimensional Markov semigroup defined in terms of Dunkl operators.
As an application of the Gagliardo-Nirenberg inequality, we prove an existence and uniqueness result for the Cauchy problem with small data of the nonlinear damped wave equation. Furthermore, we prove ergodicity for an infinite dimensional Markov semigroup defined in terms of Dunkl operators.
Version
Open Access
Date Issued
2019-04
Date Awarded
2019-08
Copyright Statement
Creative Commons Attribution-NonCommercial 4.0 International Licence
License URL
Advisor
Zegarlinski, Boguslaw
Laptev, Ari
Sponsor
EPSRC-Roth Scholarship
Publisher Department
Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)