[吴文俊数学中心综合报告] L^r-Helmholtz-Weyl decomposition in 3D exterior domains

It is known that in 3D exterior domains with the compact smooth boundary, two spaces X and of $L^-r$-harmonic vector fields with different boundary conditions are both of finite dimensions,  We prove that for every $L^r$-vector field u, there exists a uniquely decomposition as $u=h+ rot w+\nabla p$. On the other hand, if for the given $L^r$-vector field u we choose its harmonic part from V, then we have a similar decomposition to above, while the unique expression of u holds only for 1 < r < 3. Furthermore, the choice of p in H is determined in accordance with the threshold r = 3/2. Our result is based on the joint work with Matthias Hieber, Anton Seyferd(TU Darmstadt), Senjo Shimizu(Kyoto Univ.) and Taku Yanagisawa(Nara Women Univ.).