On distributions independent of \(x_N\) in certain non-cylindrical domains and a de Rham lemma with a non-local constraint
The author studies distributions not depending on the \(x_N\)-variable in \[ \Omega= \{(x', x_N)\in \mathbb{R}^N;\,x'\in \omega\subset \mathbb{R}^{N-1}, -D(x')< x_N< 0\}, \] where \(\omega\) is an open domain, \(D: \omega\to\mathbb{R}\), and \(D(x')> 0\) for all \(x'\in \omega\). Theorem. Assume that \(\text{essinf}_KD> 0\) for every compact set \(K\subset\omega\). Then the following assertions are equivalent: (1) \(F\in D'(\Omega)\) and \(\partial F/\partial x_N= 0\). (2) There exists a unique distribution \(S\in D'(\omega)\) such that \[ \langle F,\phi\rangle_{D'(\Omega),D(\Omega)}= \biggl\langle S,\int_{[-D(x'),0]}\phi(x', x_N)\,dx_N\biggr\rangle_{D'(\omega),D(\omega)} \] for all \(\phi\in D(\Omega)\). Next, let \(\nabla'\) be the divergence operator with respect to \(x'= (x_1,\dots, x_{N-1})\), and \[ C(\Omega):= \Biggl\{\phi\in D(\Omega)^{N- 1};\,\nabla'\cdot\biggl(\int_{[-D(x'),0]}\phi(x', x_N)\,dx_N\biggr)= 0\text{ in }\omega\Biggr\}. \] The author's de Rham-like lemma reads: Let \(F\in W^{-1,q'}(\Omega)^{N-1}\). Then the following conditions are equivalent: (1) \(\langle F,\phi\rangle= 0\) for all \(\phi\in C(\Omega)\cap W^{1,q}_0(\Omega)^{N-1}\). (2) \(F\) does not depend upon the \(x_N\)-variable and there exists \(p_s\in L^{q'}_{\text{loc}}(\omega)\) (unique up to an additive constant) such that \(F= \nabla'p_s\). This lemma with \(N= 3\) can be applied in the analysis of the hydrostatic approximation of the Navier-Stokes equations.
scientific article; zbMATH DE number 2119156
scientific article