Shape optimization for eigenvalues of higher order elliptic operators
高阶椭圆算子特征值的形状优化
基本信息
- 批准号:396521072
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Grants
- 财政年份:2017
- 资助国家:德国
- 起止时间:2016-12-31 至 2020-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
Shape optimization is a quickly growing field within geometric analysis. Formally, shape optimization problems are formulated as follows: given a set A of admissible shapes and a domain functional F, we look for an element of A which minimizes F in A. A classical example is the isoperimetric inequality, which is one of the oldest optimal shape problems. In this case, A is the set of all open sets with fixed volume and F is the perimeter of a domain. Things become more complex, if solutions of partial differential equations are involved. In this case, the functional F depends on a domain D and on the solution u of a given partial differential equation on D. Our interest is to understand problems of this type. We are led to the following question: a) Does there exist an optimal domain?b) Is the optimal domain regular?c) If it is, can we formulate necessary conditons of optimality? Is the optimal domain unique?For problems concerning elliptic partial differential equations of second order, there are several strategies known which help answering the above questions. These strategies are mainly based on the maximum principle, blow-up techniques or symmetrizations arguments (see e.g. Alt and Caffarelli). Unfortunately, these strategies only work if at most second order partial differential equations occur. However, we are interested in problems in which elliptic partial differential equations of higher order are involved. Hence, the previously mentioned strategies are not applicable. In this project, we concentrate on two special domain functionals in which solutions of fourth order partial differential equations are involved. Namely, the principal frequency and the buckling load of a clamped plate. Core of this project is to advance the recent progress on these two problems. Since the previously mentioned strategies are not applicable, the challange is to develop new methods to analyze our optimal shape problems. For the buckling load, we recently gain a significant progress on answering the questions a) and c). We will try to expand our results und transfer it to the fundamental frequency problem. Currently, for a minimizing domain for the buckling load or the principal frequency of a clamped plate, there are no regularity results known. Inspired by the Alt-Caffarelli approach for second order elliptic problems, one aim of this project is to find a method to get regularity of the optimal domain using the regularity of the associated solution.
在几何分析中,形状优化是一个迅速发展的领域。形式化地,形状优化问题是这样描述的:给定一个允许形状的集合A和一个区域泛函F,我们在A中寻找一个最小化F的元素。一个经典的例子是等周不等式,它是最古老的最优形状问题之一。在这种情况下,A是所有具有固定体积的开集的集合,F是区域的周长。如果涉及到偏微分方程解,事情就会变得更加复杂。在这种情况下,泛函F依赖于区域D和D上给定的偏微分方程解u。我们感兴趣的是理解这类问题。我们被引出如下问题:A)存在最优域吗?B)最优域是正则的吗?C)如果是,我们能给出最优性的必要条件吗?最优域是唯一的吗?对于二阶椭圆型偏微分方程的问题,有几种已知的策略可以帮助回答上述问题。这些策略主要基于最大值原理、爆破技巧或对称化论点(见Alt和Caffarelli)。不幸的是,这些策略只有在最多出现二阶偏微分方程的情况下才有效。然而,我们感兴趣的是涉及高阶椭圆型偏微分方程的问题。因此,上述战略并不适用。在这个项目中,我们集中于两个特殊的区域泛函,其中涉及到四阶偏微分方程解的问题。即固支板的主频和屈曲载荷。该项目的核心是推进这两个问题的最新进展。由于前面提到的策略不适用,挑战是开发新的方法来分析我们的最优形状问题。对于屈曲载荷,我们最近在回答问题a)和c)方面取得了重大进展。我们将尝试扩展我们的结果,并将其转移到基频问题。目前,对于固支板的屈曲载荷或主频的极小值域,尚无规律性结果。受二阶椭圆型问题的Alt-Caffarelli方法的启发,本项目的一个目的是找到一种利用相关解的正则性来获得最优域的正则性的方法。
项目成果
期刊论文数量(1)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Existence and stability of solutions for a fourth order overdetermined problem
四阶超定问题解的存在性和稳定性
- DOI:10.1016/j.jmaa.2021.125531
- 发表时间:2022
- 期刊:
- 影响因子:1.3
- 作者:A. Gilsbach;K. Stollenwerk
- 通讯作者:K. Stollenwerk
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Dr. Kathrin Stollenwerk其他文献
Dr. Kathrin Stollenwerk的其他文献
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