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.

我的建议致力于许多相互关联的问题，这些问题源于数学物理学，在现代分析的多个领域中发挥了核心作用，这些方案将导致加拿大数学家在这些领域的重大进展和领导。 I.通过单数矢量场（漂移）扰动的布朗运动是许多数学物理学模型的主要组成部分。它被构造为相应的随机微分方程（SDE）的解决方案。寻找漂移的最大可接受奇异性，即相应的SDE具有独特的解决方案，吸引了许多数学家的兴趣，但仍然远非完整。我打算通过应用新的运算符理论技术来实质性地推进临界界奇点，这些技术最近使我首次将漂移的关键点和关键点和关键的超出表面奇异性（在此问题的较弱变化中，即构建相关的偶性过程中）。接下来，我打算开发研究此类SDE的解决方案所需的工具，包括（非高斯）双侧边界，对相应的Kolmogorov向后操作员的基本解决方案。 ii。我继续致力于解决Schroedinger操作员在r^d上缺乏正面特征值的长期问题，在尺寸d = 3或更高的情况下，以及针对Schroedinger操作员特征函数的独特延续（UC）的相关问题。我打算通过利用使用与UC链接的运算符理论技术（扩展了我与L. shartser的链接）以及一种不依赖UC（一种新方法）来获得有关缺乏积极特征值的新结果，该技术缺乏积极的特征值。 I和II项目的目标是将现代运营商理论技术带入差异过程和独特的延续领域。 iii。最近，I（与A. brudnyi共同）通过将Cartan定理A和B（Oka-Cartan理论）扩展到同时的型号和相干型套管上的各种示例（几乎是个性化的范围），从而在Stein歧管的某些代数内建立了复杂功能理论的基本结果。物理，例如安德森本地化）。这项工作表明，Oka-Cartan理论是研究D-BAR方程的复杂功能理论的一种方法，超出了复杂歧管的经典设置。我打算将开发的技术扩展到具有相似的本地结构但具有不同的全球结构的霍明型函数代数，例如hardy代数的某些子代数（在多核方面获得这些代数的电晕理论），旨在确定Oka-Cartan理论的“自然领域”。