Abstract
The contents of this thesis are an assortment of results in analysis and subRiemannian geometry, with a special focus on the Heisenberg group Hn, Heisenbergtype (H-type) groups, and Carnot groups.
As we wish for this thesis to be relatively self-contained, the main definitions and background are covered in Chapter 1. This includes basic information about Carnot groups, Hn, H-type groups, diffusion operators, and the curvature dimension inequality.
Chapter 2 incorporates excerpts from a paper by N. Garofalo and the author, [42]. In it, we propose a generalization of Almgren’s frequency function N : (0, 1) → R for solutions to the sub-elliptic Laplace equation ΔHu = 0 in the unit ball of a Carnot group of arbitrary step. If the function u has vanishing discrepancy, then the frequency is monotonically non-decreasing, and we are able to prove a form of strong unique continuation for such functions.
Chapter 3 grew out of the author seeking parabolic montonicity formulas in the same vein as Almgren’s frequency. These include two types of monotonicity formulas, those of Struwe- and Poon-type [72], [67]. If a diffusion operator L on a complete manifold M satisfies the curvature dimension inequality CD(ρ, n), then we are able to prove that for solutions to L u = ut in M × (0, T), Struwe’s energy monotonicity holds, at least for time values close enough to T. We introduce a new condition, C(ω) where ω ∈ C1(0, T), related to the Hessian of the heat kernel, and are able to prove a Poon-type frequency monotonicity formula when taking into account a weighting factor depending on ω. We also give examples of manifolds satisfying C(ω), the most interesting of which includes the Ornstein-Uhlenbeck operator. Monotonicity of the weighted frequency also implies a form of strong-unique continuation.
In Chapter 4, we derive asymptotics for the heat kernel on H-type groups and generalize a gradient bound from a paper of Garofalo and Segala [43] to these groups. This gradient bound in turn implies a strong Harnack inequality and Wiener criterion similar to those found in [31] and [43].
Keywords
Pure sciences, Carnot groups, Heat kernel, Monotonicity formulas, Unique continuation, Wiener's criterion
Disciplines
Mathematics
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Department
Mathematics
First Advisor
Nicola Garofalo
Second Advisor
Donatella Danielli
Committee Chair
Nicola Garofalo
Committee Co-Chair
Donatella Danielli
Committee Member 1
Fabrice Baudoin
Committee Member 2
Nung Kwan Yip
Date of Award
4-2016
Recommended Citation
Rotz, Kevin L., "Monotonicity Formulas for Diffusion Operators on Manifolds and Carnot Groups, Heat Kernel Asymptotics and Wiener's Criterion on Heisenberg-type Groups" (2016). Open Access Dissertations. 700.
https://docs.lib.purdue.edu/open_access_dissertations/700