Monotonicity Formulas for Diffusion Operators on Manifolds and Carnot Groups, Heat Kernel Asymptotics and Wiener's Criterion on Heisenberg-type Groups

<p>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.</p> <p>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.</p> <p>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 <em>N</em> : (0, 1) → R for solutions to the sub-elliptic Laplace equation Δ<em>Hu</em> = 0 in the unit ball of a Carnot group of arbitrary step. If the function <em>u</em> 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.</p> <p>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 <em>L</em> on a complete manifold M satisfies the curvature dimension inequality CD(ρ, <em>n</em>), then we are able to prove that for solutions to <em>L</em><em> u</em> = <em>u</em>t in M × (0, <em>T</em>), Struwe’s energy monotonicity holds, at least for time values close enough to <em>T.</em> We introduce a new condition, <em>C</em>(ω) where ω ∈ <em>C</em>1(0,<em> T</em>), 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 <em>C</em>(ω), the most interesting of which includes the Ornstein-Uhlenbeck operator. Monotonicity of the weighted frequency also implies a form of strong-unique continuation.</p> <p>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].</p>

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Year

2016

Author dc:creator

• Rotz, Kevin L

Contributors dc:contributor

• Nicola Garofalo
• Donatella Danielli
• Fabrice Baudoin
• Nung Kwan Yip

• Pure sciences
• Carnot groups
• Heat kernel
• Monotonicity formulas
• Unique continuation
• Wiener's criterion
• Mathematics

Repository record dc:identifier

https://docs.lib.purdue.edu/open_access_dissertations/700

OAI identifier oai:identifier

oai:docs.lib.purdue.edu:open_access_dissertations-1848