Construction of bases in spaces of solenoidal vector fields (Q937944)

The author considers the classical problem \[ \text{div}\,w=\psi\quad\text{in}\;\Omega,\quad w| _{\partial\Omega}=0,\quad \int\limits_\Omega \psi\,dx=0,\tag{1} \] where \(\Omega\subset \mathbb{R}^n\) is a star-shaped bounded domain. It is proved that the problem (1) has a solution given by a certain potential and this solution admits the estimate \[ \|\nabla w\|_{q,\Omega}\leq\beta(n,q,d)\|\psi\|_{q,\Omega}, \] where \(d=\text{dist}\{\partial\Omega;0\}\). On the base of this result a method of construction of fundamental systems in the space \(H(\Omega)\) of solenoidal vector fields is described. Namely, an operator \(P:\,\overset \circ W^1_2(\Omega)\rightarrow H(\Omega)\) is constructed such that if \(\{\varphi_k\}\) is a fundamental system in \(\overset \circ W^1_2(\Omega)\) then \(\{P(\varphi_k)\}\) is a fundamental system in \(H(\Omega)\).