Mathematical Sciences: Semi-Linear Elliptic Problems

本课题是研究偏微分方程的稳态问题。主要研究半线性椭圆问题解的存在性、多重性和对称性问题。该项目由两部分组成。在第一部分中，主要研究者考虑了涉及参数的半线性问题，并研究了问题的线性部分的特征值的一个邻域内参数的解结构。这里的主要研究者主要对非线性部分涉及有界振荡项的半线性问题（正弦戈登型问题）感兴趣。对于这类问题，基态（正解）的多重性似乎与自变量的维数和底层域的几何形状密切相关。事实上，我们对非凸域所知甚少。第二部分主要研究了半线性椭圆型问题正解分支的对称破缺问题和内层边界层的发展。这里的主要研究者主要对具有一定反射对称性的球面或凸域的问题感兴趣。

This project is concerned with the study of the steady states of partial differential equations. In particular, the principal investigator is interested in existence, multiplicity and symmetry questions for solutions of semilinear elliptic problems. The project consists of two parts. In the first part the principal investigator considers semilinear problems involving a parameter and studies the solution structure of the problem for the parameter in a neighborhood of an eigenvalue of the linear part of the problem. Here the principal investigator is mainly interested in semilinear problems whose nonlinear part involves bounded oscillatory terms (Sine Gordon type problems). For such problems the multiplicity of ground states (positive solutions) appears intimately dependent upon the dimension of the independent variable and the geometry of the underlying domain. In fact, very little is known for domains which are not convex. In the second part the principal investigator studies the question of symmetry breaking and the development of interior and boundary layers for positive solution branches of semilinear elliptic problems. Here the principal investigator is mainly interested in problems where the domain is either a sphere or a convex domain with certain reflection symmetries.