由本体和曲面偏微分方程组成的二阶和四阶椭圆系统:适定性、正则性理论和特征值问题

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2020-08-03 DOI:10.4171/IFB/463

P. Knopf, Chun Liu

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引用次数: 8

摘要

本文研究了二阶和四阶椭圆型问题，这两类问题不仅在整体上包含泊松方程，而且在区域边界上包含非齐次拉普拉斯-贝尔特拉米方程。体积和表面偏微分方程由狄利克雷或罗宾型边界条件耦合。我们指出Dirichlet型和Robin型边界条件可以通过我们的形式同时处理，而不必改变框架。此外，我们研究了与这些二阶和四阶椭圆系统相关的特征值问题。进一步讨论了这些椭圆型问题与某些抛物型问题的关系，特别是具有动态边界条件的Allen—Cahn方程和Cahn—Hilliard方程。

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On second-order and fourth-order elliptic systems consisting of bulk and surface PDEs: Well-posedness, regularity theory and eigenvalue problems

In this paper, we study second-order and fourth-order elliptic problems which include not only a Poisson equation in the bulk but also an inhomogeneous Laplace--Beltrami equation on the boundary of the domain. The bulk and the surface PDE are coupled by a boundary condition that is either of Dirichlet or Robin type. We point out that both the Dirichlet and the Robin type boundary condition can be handled simultaneously through our formalism without having to change the framework. Moreover, we investigate the eigenvalue problems associated with these second-order and fourth-order elliptic systems. We further discuss the relation between these elliptic problems and certain parabolic problems, especially the Allen--Cahn equation and the Cahn--Hilliard equation with dynamic boundary conditions.

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.