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 $[L^r(\Omega)]^d, 1<r<\infty.$ 
   We will give a few remarks on the application of these results to the 
solutions of Maxwell's equations with suitable boundary conditions.

Reference:Hermann Ludwig Ferdinand von Helmholtz (Aug 31, 1821--Sep 8, 1894),
"\"Uber Integrale der hydrodynamischen Gleichungen, welcher der 
Wirbelbewegungen entsprechen" (On integrals of the hydrodynamic equations 
which correspond to vortex motions), Journal f\"ur die reine und angewandte 
Mathematik, 55: 25--55  (1858)