$L^r$-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains Hideo KozonoTaku Yanagisawa 35Q30$L^r$-vector fieldsharmonic vector fieldsBetti numberdiv-curl lemma We show that every $L^{r}$-vector field on $\Omega$ can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators $\text{rot}$ and $\mathop{div}$, where $\Omega$ is a bounded domain in $\mathbb{R}^3$ with the smooth boundary $\partial\Omega$. Our decomposition consists of two kinds of boundary conditions such as $u \cdot \nu|_{\partial\Omega} = 0$ and $u \times \nu|_{\partial\Omega} = 0$, where $\nu$ denotes the unit outward normal to $\partial\Omega$. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of $C^{\infty}$-forms on compact Riemannian manifolds into $L^{r}$-vector fields on $\Omega$. As an application, the generalized Biot-Savart law for the incompressible fluids in $\Omega$ is obtained. Furthermore, various bounds of $u$ in $L^{r}$ for higher derivatives are given by means of $\text{rot} u$ and $\mathop{div} u$. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3605 10.1512/iumj.2009.58.3605 en Indiana Univ. Math. J. 58 (2009) 1853 - 1920 state-of-the-art mathematics http://iumj.org/access/