3月16日(金) 13:30 -- 14:30 申 東雨 (大韓民国, ソウル国立大学) ``The Helmholtz and Leray decompositions and their applications to Maxwell's equations.'' <研究発表要旨> In 1858 Hermann von Helmholtz introduced the idea of decomposing a fluid vector field $(u,v,w)$ as a sum of $\nabla p + \nabla\times (L,M,N).$ Since then, the theory of Helmholtz decomposition of vector field has been elaborated by many mathematicians and now it becomes one of the most fundamental and useful theory in the analysis of solutions of partial differential equations arising from fluid/solid mechanics and electromagnetics. In this talk we will review recent results on the Helmholtz and Leray decompositions for vector fields. Under various assumptions on $\Omega\in \mathbb R^d,$ for $d =2,3,$ we will first look at the classical decomposition theory of $[L^2(\Omega)]^d$, and then the corresponding theory of $[L^r(\Omega)]^d, 1