Operator-theoretic approach to problems of Analysis and Partial Differential Equations

分析和偏微分方程问题的算子理论方法

• 批准号: RGPIN-2017-05567
• 负责人: Kinzebulatov, Damir
• 金额: $ 1.53万
• 依托单位: Université Laval
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711258
• 关键词: Operator; theoretic; approach; problems; Analysis

My proposal is devoted to a number of interrelated problems originating in Mathematical Physics, playing a central role in several areas of modern Analysis, whose solution would lead to a significant progress and leadership of Canadian mathematicians in these areas. I. A Brownian motion perturbed by a singular vector field (drift) is the principal component of many models of Mathematical Physics. It is constructed as a solution of the corresponding stochastic differential equation (SDE). The search for the maximal admissible singularities of the drift, i.e. such that the corresponding SDE has a unique solution, attracted the interest of many mathematicians, but is still far from being complete. I intend to substantially advance this search, reaching critical-order singularities, by applying new operator-theoretic techniques that recently allowed me to combine, for the first time, critical point and critical hypersurface singularities of the drift (in a weaker variant of this problem, i.e. constructing an associated Feller process). Next, I intend to develop the instruments needed to study solutions of such SDEs, including (non-Gaussian) two-sided bounds on the fundamental solution of the corresponding Kolmogorov backward operator. II. I continue to work towards solving the long-standing problem of absence of positive eigenvalues of Schroedinger operators on R^d, in dimension d=3 or higher, and related problem of unique continuation (UC) for eigenfunctions of Schroedinger operators. I intend to obtain new, close-to-optimal results on the problem of absence of positive eigenvalues by exploiting an operator-theoretic technique that uses the link to the UC (extending my earlier work with L. Shartser), and a technique that does not rely on the UC (a new approach). The goal of Projects I and II is to bring modern operator-theoretic techniques to the areas of diffusion processes and unique continuation. III. Recently, I (jointly with A. Brudnyi) established the basic results of complex function theory within certain Fréchet algebras of holomorphic functions on coverings of Stein manifolds by extending Cartan theorems A and B (Oka-Cartan theory) to coherent-type sheaves on the spectra of these algebras (model example: holomorphic almost periodic functions, arising in various problems of Analysis and Mathematical Physics, e.g. in Anderson localization). This work suggests that the Oka-Cartan theory, as an approach to complex function theory alternative to studying the d-bar equation, is valid beyond the classical setup of complex manifolds. I intend to extend the developed techniques to the algebras of holomorphic functions that have, in a sense, a similar local structure, but a different global structure, e.g. certain subalgebras of Hardy algebra on polydisk (obtaining a corona theorem for these algebras), aiming at determining the "natural domain" of Oka-Cartan theory.

我的建议是致力于一些相互关联的问题起源于数学物理，发挥了核心作用，在几个领域的现代分析，其解决方案将导致一个显着的进步和领导加拿大数学家在这些领域。 I.布朗运动受奇异向量场（漂移）的扰动是许多数学物理模型的主要组成部分。它被构造为相应的随机微分方程（ε）的解。寻找漂移的最大可容许奇点，即使得相应的漂移具有唯一解，吸引了许多数学家的兴趣，但仍然远远没有完成。我打算通过应用新的算子理论技术来大大推进这一搜索，达到临界阶奇点，这些技术最近使我能够首次将漂移的临界点和临界超曲面奇点（在这个问题的一个较弱的变体中，即构建一个相关的费勒过程）联合收割机。接下来，我打算开发的工具，研究解决方案，这样的SDES，包括（非高斯）双边边界上的基本解决方案的相应的柯尔莫哥洛夫向后运营商。 二.我继续致力于解决长期存在的问题的缺乏正特征值的薛定谔算子在R^d，在d=3或更高的维度，和相关的问题的唯一延续（UC）的本征函数的薛定谔算子。我打算利用算子理论的技术，使用UC的链接（扩展我以前的工作与L。Shartser），以及一种不依赖于UC的技术（一种新方法）。 项目I和II的目标是将现代算子理论技术带到扩散过程和独特延续的领域。 三.最近，我（与A。Brudnyi）通过将Cartan定理A和B（Oka-Cartan理论）扩展到这些代数的谱上的相干型层（模型例子：全纯概周期函数，出现在分析和数学物理的各种问题中，例如在安德森局部化中），建立了Stein流形覆盖上的全纯函数的某些Fréchet代数内的复变函数理论的基本结果。这项工作表明，奥卡-嘉当理论，作为一种方法来研究的d-杆方程的替代复函数理论，是有效的，超越了经典的设置复杂的流形。我打算将已开发的技术推广到全纯函数的代数，在某种意义上，有一个类似的局部结构，但不同的全球结构，例如某些子代数的多圆盘上的哈代代数（获得电晕定理，这些代数），旨在确定“自然域”的奥卡-嘉当理论。