Approximation of potential and solenoidal vector fields
The approximation of solenoidal vector fields in the Sobolev space \(W_ p^{\ell}(\Omega)\) and of potential vector fields in \(L_ p(\Omega)\) by smooth vector fields is studied. Let \(G_ p(\Omega)\) be the closure in \(L_ p(\Omega)\) of the space of restrictions on \(\Omega\) of potential vector fields v(x)\(\in \{\nabla \psi (x):\psi (x)\in C_ 0^{\infty}({\mathbb{R}}^ n)\}\) \((v(x)=(v_ 1,...,v_ n),x=(x_ 1,...,x_ n)\), \(\Omega \subset {\mathbb{R}}^ n\), \(n\geq 2\), \(1<p<\infty)\), \(\hat G_ p(\Omega)=\{v(x)=\nabla \psi (x)\in L_ p(\Omega)\}\), \(\overset\circ{\mathcal F}^{\infty}(\Omega)=\{v(x)\in C_ 0^{\infty}(\Omega):div v=0\}\), \(\overset\circ {\mathcal F}_ p^{\ell}(\Omega)\) the closure of the solenoidal vector fields \(\overset\circ{\mathcal F}^{\infty}(\Omega)\) in the space \(\overset\circ W_ p^{\ell}(\Omega)\) of n-dimensional vector fields and \(\overset\circ{\hat {\mathcal F}}_ p^{\ell}(\Omega)=\{v(x)\in \overset\circ W_ p^{\ell}(\Omega)\); div v\(=0\}\) (\(\ell \geq 0\) an integer). It is proved that for any domain \(\Omega\) the equality (*) \(\overset\circ{ \hat {\mathcal F}}_ p^{\ell}(\Omega)=\overset\circ {\mathcal F}_ p^{\ell}(\Omega)\) is equivalent to \(\hat G_{p'}(\Omega)=G_{p'}(\Omega)\) \((p'=p/(p-1))\). Among other things the following results are proved. \({\overset\circ {\mathbb{F}}}_ p^{\ell}(\Omega)=\overset\circ {\mathcal F}_ p^{\ell}(\Omega)\) holds: (1) for bounded Lipschitz domains with a compact boundary \((1\leq p<\infty),\) (2) for an arbitrary unbounded Lipschitz domain, if \(1\leq p\leq n/(n-1)\). Also, for the case \(n/(n-1)<p<\infty\), sufficient conditions for the bounded Lipschitz domain \(\Omega\) with noncompact boundary are given such that (*) holds.
approximation of solenoidal vector fields
bounded Lipschitz domains with a compact boundary
scientific article; zbMATH DE number 3967096