Mathematical Sciences: Degenerate Microlocal Methods in Geometric Analysis

这个建议是在几何和微局部分析的一般领域。研究者打算继续发展退化伪微分技术，并将其应用于椭圆型偏微分方程的散射理论和谱分析。这些技术包括参数的显式几何构造和退化椭圆算子的广义逆。物理和几何中的许多问题都可以用偏微分方程组来描述。椭圆型偏微分方程——例如，描述各种热流现象——在偏微分方程理论中形成了一个特别发达的分支。提出的研究涉及椭圆型微分方程的“奇异”方面，这在理论上比光滑情况更困难。

9626382 Mazzeo This proposal is in the general area of geometric and microlocal analysis. The investigator intends to continue the development of degenerate pseudodifferential techniques and to apply them to scattering theory and spectral analysis of elliptic partial differential equations. These techniques involve explicit geometric constructions of parametrics and generalized inverses for degenerate elliptic operators. Many problems in physics and geometry can be described by systems of partial differential equations. Elliptic partial differential equations - describing, for example, various heat flow phenomena - form a particularly well-developed branch within the theory of partial differential equations. The proposed research deals with the 'singular' aspect of elliptic differential equations which are theoretically more difficult than the smooth case.