Mathematical Sciences: On Nonlinear Elliptic Equations

这个项目的工作扩展了早先对半线性椭圆偏微分方程理论的贡献。近年来的研究表明，当方程在无界区域上定义时，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，以保证任何解的存在。简单的例子说明一个方程可能只有平凡的解。第二项研究将集中在二阶全非线性椭圆方程上。这是一个相当活跃的领域，已经建立了非负截面曲率紧致黎曼流形上方程的存在性和唯一性结果。目前尚不清楚非否定性是否必要。我们将努力解释截面曲率的作用，并把相关的复杂方程作为一个更长远的目标来考虑。

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.