On an overdetermined problem for composite materials

复合材料的超定问题

• 批准号: 22K13935
• 负责人: Cavallina Lorenzo
• 金额: $ 2.91万
• 依托单位: Tohoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Early-Career Scientists
• 财政年份: 2022
• 资助国家: 日本
• 起止时间: 2022-04-01 至 2026-03-31
• 项目状态: 未结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-22K13935/
• 关键词: 優決定問題; 形状間関数; 陰関数定理; 楕円型偏微分方程式; 分岐解析; 二相問題; 対称性; 形状最適化問題; 二相

本年度報告する主な研究成果は以下のものである.① 論文[C., G. Poggesi, T. Yachimura, Nonlinear Anal. 2022] では, Serrin 型優決定問題において, 一相と二相の場合を比較して, 定量的な安定性の評価を与えた. 一相の場合は, 本優決定問題の解が球に限る[Serrin 1971]. 一方で, 二相の場合は非自明な解もまた存在することが知られている[C.-Yachimura2020]. 本論文では, 二相Serrin 型優決定問題の問題設定が「一相に近い」という仮定の下で, 二相Serrin 型優決定問題の解は「球に近い」という定量的な評価を与えた.② 論文[C.,T. Yachimura, Current Trends in Analysis 2022]では, [C.-Yachimura 2020] で扱えなかった「退化な設定」（伝導率がcritical value のときに相当する）において, 二相Serrin型優決定問題の解の枝が分岐することを示した.③ 論文[C., Indiana Univ. Math. J. 2022] では, 「境界上の法線微分の値が平均曲率と比例する」という条件を満たす二相楕円型方程式における優決定問題を考えた. 優決定条件に現れる比例定数によって, 「A) 解は存在しない」, 「B) 解は自明解 (同心球) に限る」, 「C) 自明解に加えて、非自明解もまた存在する」の三パターンの特徴づけを行った. また, C) の場合, 分岐解析を行った.

In this year's report, the main research results are as follows: article 1 [C., G. Poggesi, T. Yachimura, Nonlinear Anal. 2022] the Serrin type determines the problem, one phase, two phases, quantitative stability and stability. As soon as we meet each other, we decide to solve the problem and the limit of the ball [Serrin 1971]. On the one hand, the combination of the two phases is not self-evident. There is a problem that you know that you have a problem [C.-Yachimura2020]. In this paper, the two-phase Serrin type decision problem setting, "one-phase proximity", the two-phase Serrin type decision problem solution, "spherical proximity", the quantitative comparison and comparison of the two-phase Serrin type decision problem. 2 [C. Yachimura, Current Trends in Analysis 2022] [C.-Yachimura 2020] the two-phase serial model determines the problem of solving branch bifurcation problems. 3. The paper [C., Indiana Univ. Math. J. 2022] in the realm, the normal differential equation, the mean curvature ratio, the boundary condition, the two-phase equation, the equation. To determine the condition, the ratio is determined, "A) the solution is limited," B) the solution is self-explanatory (concentric sphere), and "C) it is self-explanatory and non-self-explanatory. The lines are closed and the bifurcations are resolved.

项目成果

Local analysis of a two phase free boundary problem concerning mean curvature

涉及平均曲率的两相自由边界问题的局部分析

• DOI: 10.1512/iumj.2022.71.9014
• 发表时间: 2022
• 期刊: Indiana University Mathematics Journal
• 影响因子: 1.1
• 作者: Lorenzo Cavallina;Giorgio Poggesi;Toshiaki Yachimura;Cavallina Lorenzo
• 通讯作者: Cavallina Lorenzo

On an overdetermined problem of Serrin-type in a two-phase composite medium with imperfect interfaces

不完善界面两相复合介质中Serrin型超定问题

• 发表时间: 2023
• 作者: Cavallina Lorenzo;Yachimura Toshiaki;Cavallina Lorenzo
• 通讯作者: Cavallina Lorenzo

Quantitative stability estimates for a two-phase Serrin-type overdetermined problem

两相 Serrin 型超定问题的定量稳定性估计

• DOI: 10.1016/j.na.2022.112919
• 发表时间: 2022
• 期刊: Nonlinear Analysis
• 作者: Lorenzo Cavallina;Giorgio Poggesi;Toshiaki Yachimura
• 通讯作者: Toshiaki Yachimura

形状汎関数の非退化な臨界点における対称性と非対称性について

关于形泛函非简并临界点的对称性和不对称性

• 发表时间: 2022
• 作者: Cavallina Lorenzo;Yachimura Toshiaki;Cavallina Lorenzo;Cavallina Lorenzo;Cavallina Lorenzo
• 通讯作者: Cavallina Lorenzo

Symmetry Breaking Solutions for a Two-Phase Overdetermined Problem of Serrin-Type

Serrin型两相超定问题的对称破缺解

• DOI: 10.1007/978-3-030-87502-2_44
• 发表时间: 2022
• 期刊: Current Trends in Analysis, its Applications and Computation Proceedings of the 12th ISAAC Congress, Aveiro, Portugal, 2019
• 作者: Cavallina Lorenzo;Yachimura Toshiaki
• 通讯作者: Yachimura Toshiaki