\(L^r\) -Helmholtz–Weyl Decomposition in Three Dimensional Exterior Domains

This article is a survey of our papers (Hieber et al. in J Funct Anal 281:109144, 52, 2021, [8]) and (Hieber et al. in J Geom Anal 32:206, 26, 2022, [10]). In 3D exterior domains \(\Omega \) with the compact smooth boundary \(\partial \Omega \) , two spaces \(X^r_{\tiny {\text{ har }}}(\Omega )\) and \(V^r_{\tiny {\text{ har }}}(\Omega )\) of \(L^r\) -harmonic vector fields \(\boldsymbol{h}\) with \(\boldsymbol{h}\cdot \boldsymbol{\nu }|_{\partial \Omega } =0\) and \(\boldsymbol{h}\times \boldsymbol{\nu }|_{\partial \Omega } =\boldsymbol{0}\) are both of finite dimensions, where \(\boldsymbol{\nu }\) denotes the unit outward normal to \(\partial \Omega \) . For every \(L^r\) -vector field \(\boldsymbol{u}\) , there exist \(\boldsymbol{h} \in X^r_{\tiny {\text{ har }}}(\Omega )\) , \(\boldsymbol{w}\in \dot{H}^{1,r}(\Omega )^3\) with \(\hbox {div }\boldsymbol{w} =0\) and \(p \in \dot{H}^{1,r}(\Omega )\) such that \(\boldsymbol{u}\) is uniquely decomposed as \(\boldsymbol{u} = \boldsymbol{h} + \hbox {rot }\boldsymbol{w} + \nabla p\) . The corresponding result to bounded domains has been obtained in Kozono and Yanagisawa (Indiana Univ Math J 58:1853–1920, 2009, [13]). On the other hand, if for the given \(L^r\) -vector field \(\boldsymbol{u}\) we choose its harmonic part \(\boldsymbol{h}\) from \(V^r_{\tiny {\text{ har }}}(\Omega )\) , then we have a similar decomposition to above, while the unique expression of \(\boldsymbol{u}\) holds only for \(1 < r < 3\) . Furthermore, the choice of p in \(\dot{H}^{1, r}(\Omega )\) is determined in accordance with the threshold \(r = 3/2\) . Our method relies on the \(L^r\) -variational inequality which enables us to construct \(\boldsymbol{w} \in \dot{H}^{1, r}(\Omega )^3\) and \(p\in \dot{H}^{1, r}(\Omega )\) for given \(\boldsymbol{u} \in L^r(\Omega )^3\) as weak solutions to the elliptic boundary value problems.