Nodal geometry of eigenfunctions on smooth manifolds and hardy-littlewood-sobolev inequalities on the heisenberg group

<p>Part I: Let (M,g) be a n dimensional smooth, compact, and connected Riemannian manifold without boundary, consider the partial differential equation on M:</p> <p>-Δu=Λu,</p> <p>in which Δ is the Laplace-Beltrami operator. That is, u is an eigenfunction with eigenvalue Λ. We analyze the asymptotic behavior of eigenfunctions as Λ go to ∞ (i.e., limit of high energy states) in terms of the following aspects.</p> <p>(1) Local and global properties of eigenfunctions, including several crucial estimates for further investigation.</p> <p>(2) Write the nodal set of u as N={u=0}, estimate the size of N using Hausdorff measure. Particularly, surrounding the conjecture that the n-1 dimensional Hausdorff measure is comparable to square root of Λ, we discuss separately on lower bounds and upper bounds.</p> <p>(3) BMO (bounded mean oscillation) estimates of eigenfunctions, and local geometric estimates of nodal domains (connected components of nonzero region).</p> <p>(4) A covering lemma which is used in the above estimates, it is of independent interest, and we also propose a conjecture concerning its sharp version.</p> <p>Part II: O the Heisenberg group with homogeneous dimension Q=2n+2, we study the Hardy-Littlewood-Sobolev (HLS) inequality,</p> <p>and particularly its sharp version. Weighted Hardy-Littlewood-Sobolev inequalities with different weights shall also be investigated, and we solve the following problems.</p> <p>(1) Establish the existence results of maximizers.</p> <p>(2) Provide a upper bound of sharp constants.</p>

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Open Access Dissertation

Discipline thesis:degree_discipline

Mathematics

Year dc:date.available

2012

Author dc:creator

• Han, Xiaolong

Contributors dc:contributor

• Guozhen Lu

• eigenfunctions of Laplacian on smooth manifolds, existence of maximizers, geometric estimates of nodal sets and nodal domains, Hardy-Littlewood-Sobolev inequalities, Heisenberg group
• Mathematics

Repository record dc:identifier

https://digitalcommons.wayne.edu/oa_dissertations/507

OAI identifier oai:identifier

oai:digitalcommons.wayne.edu:oa_dissertations-1506