Variational problems related to the 1-Laplace operator

与 1-拉普拉斯算子相关的变分问题

• 批准号: 117345648
• 负责人: Professor Dr. Friedemann Schuricht
• 金额: --
• 依托单位: Institut für Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2009
• 资助国家: 德国
• 起止时间: 2008-12-31 至 2012-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/117345648?language=en
• 关键词: Variational; problems; related; Laplace; operator

Various problems in nature, technology and economy lead to a so called variational problem where a certain energy function has to be minimized under suitable constraints and the solution satisfies some associated differential equation. The methods for the treatment of such problems are mainly based on the differentiability of the involved functionals and operators. However, in the recent applications one is more and more confronted with situations where the usual assumptions concerning differentiability are not available. However, the study of such nondifferentiable problems is still in an early stage. The present project intends to contribute to this development by investigating current issues occuring in the context of the 1-Laplace operator that enjoys increasing interest for instance in image processing. The comprehensive experience of the applicant in the treatment of nondifferentiable problems in mechanics and geometry and, recently, also in the problems under consideration are certainly a solid foundation for successful investigations. The project aims to contribute to the treatment of a permanently growing class of problems in calculus, geometry, and applications, which will certainly gain significance in the near future, not only by the investigation of concrete problems but also by the development and establishment of methods.

自然界、技术和经济中的各种问题都会导致所谓的变分问题，即在适当的约束条件下最小化某个能量函数，其解满足相关的微分方程。处理这类问题的方法主要基于所涉及的泛函和算子的可微性。然而，在最近的应用中，人们越来越多地遇到关于可微性的通常假设不成立的情况。然而，这类不可微问题的研究还处于初级阶段。目前的项目打算通过调查在1-拉普拉斯算子的背景下发生的当前问题来促进这一发展，例如在图像处理方面。申请人在处理力学和几何中的不可微问题以及最近正在考虑的问题方面的综合经验无疑是成功研究的坚实基础。该项目旨在为处理微积分、几何和应用中不断增长的一类问题做出贡献，这在不久的将来肯定会获得重要意义，不仅是通过对具体问题的研究，而且是通过方法的发展和建立。