Continuity properties of the inf-sup constant for the divergence (Q2798648)

The stable solvability and uniqueness of the solutions of a slow Stokes flow in a domain \(D\) of \(\mathbb R^3 \) are related with the following ``inf-sup condition'': NEWLINE\[NEWLINE \exists \beta(D) >0 \text{ s.t. }\inf _{p \in V} \sup_{u \in W} \{ (\operatorname{div}\, u, p) / \| u\|\,\| p\| \} \geq \beta(D),NEWLINE\]NEWLINE where \(V,\) \(W\) are the corresponding Lebesgue and Sobolev spaces associated to \(D\). Moreover, \(\beta(D)\) is important in the iterative methods for solving the Navier-Stokes equations and is also known as the Ladyzhenskaya-Babuska-Brezzi constant. Explicit values of \(\beta(D)\) are known only for few domains. The present paper is an important step forward for better understanding the dependence of \(\beta\) in terms of \(D\). The main point is the study of the convergence of \(\beta_n = \beta(D_n)\) where \(D_n\) is an ``approximation'' of \(D\). \(\beta_n\) converges to \(\beta(D)\) only for smooth enough domains, and some counterexamples are given in the paper. The upper semicontinuity of \(\beta(D)\) in terms of some function spaces and some interior approximations of \(D\) is proved. The fractional Sobolev spaces are used as an important mathematical tool.