Mathematical Sciences: Semi-Linear Elliptic Problems

本课题研究的是稳态 偏微分方程的状态 特别是 主要研究者感兴趣的是存在性、多样性 半线性椭圆方程组解的性质和对称性问题 问题 该项目包括两个部分。 第一部分 主要研究者认为半线性问题 涉及一个参数，并研究了解决方案的结构， 的特征值的邻域中的参数的问题 问题的线性部分。 这里的首席研究员 主要研究半线性问题， 涉及有界振荡项（Sine Gordon型问题）。 对于这样的问题，基态的多重性（正 解决方案）似乎密切依赖于尺寸的 自变量和基础域的几何形状。 事实上，对于非凸域，我们知之甚少。 在第二部分中，主要研究者研究了 对称性破缺的问题和发展的内部和 正解分支的边界层 椭圆问题 在这里，首席研究员主要是 感兴趣的问题，域是一个球体或 具有一定反射对称性的凸域。

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.