Mathematical Sciences: On Nonlinear Elliptic Equations

这个项目的工作扩展了对半线性椭圆型偏微分方程组理论的早期贡献。最近的研究表明，当方程定义在无界区域上时，可以存在多个解。在这个项目中要考虑的问题之一是澄清必须对方程施加的条件才能保证任何解的存在。简单的例子说明了一个方程可能只有平凡的解。第二方面的研究将集中在二阶完全非线性椭圆型方程上。这是一个相当活跃的领域，在这里已经建立了具有非负截面曲率的紧致黎曼流形上的方程的存在唯一性结果。目前还不清楚这种非负性是否有必要。将努力解释截面曲率的作用，并将相关的复杂方程视为较长期的目标。

Work on this project extends earlier contributions to the theory of semilinear elliptic partial differential equations. Recent research has established that multiple solutions can exist when the equations are defined on unbounded domains. One of the issues to be considered in this project is to clarify the conditions one must impose on the equations to guarantee the existence of any solutions. Simple examples illustrate how an equation may have only trivial solutions. A second line of investigation will focus on second order fully nonlinear elliptic equations. This is an area of considerable activity where existence and uniqueness results have been established for equations on compact Riemannian manifolds with nonnegative sectional curvature. It is not clear whether the nonnegativity is necessary. Work will be done in an effort to explain the role of the sectional curvature and to consider the related complex equations as a longer-range goal.