Mathematical Sciences: Semilinear Elliptic Equations

这个项目将考虑具有临界指数的半线性偏微分方程。研究的主题包括Dirichlet边界条件下有界区域的Kazdan-Warner问题，进一步分析Sobolev不等式中的常数，以及Neumann问题解的多重性。这些问题是由不同领域的问题引起的，如微分几何和数学生物学。基本的问题，如曲面的存在与规定的曲率或性质的平衡的生物种群模型将受益于这个项目的结果。

This project will consider semilinear partial differential equations with critical exponents. Topics to be studied include the Kazdan-Warner problem in bounded regions with Dirichlet boundary condition, further analysis of the constant in the Sobolev inequality, and multiplicity of solutions for the Neumann problem. These problems are motivated by questions arising in areas as diverse as differential geometry and mathematical biology. Basic questions such as the existence of surfaces with prescribed curvature or the properties of the equilibria of biological population models will benefit from the results of this project.