On inequalities of Friedrichs and Babuška-Aziz in dimension three
The author extends Friedrichs inequality for conjugate harmonic functions to functions of three independent variables replacing conjugate harmonic functions by pairs of functions \(p\) and vector-valued functions \(q= (q_1,q_2,q_3)\) satisfying the equations \[ \text{rot }q= -\nabla p\quad\text{and}\quad \text{div }q= 0.\tag{1} \] It is shown that for a bounded and simply-connected open domain \(\Omega\) with boundary in the class \(C^2\) there exist constants \(\Gamma\) and \(\widetilde\Gamma\) depending only on the shape of \(\Omega\) such that for any pairs of functions \(p\), \(q\) satisfying (1) in the distribution sense the following inequalities hold: \[ \int_\Omega p^2dV\leq \Gamma \int_\Omega|q|^2dV\quad\text{provided that }\int_\Omega p dV= 0, \] and \[ \int_\Omega|q|^2dV\leq \widetilde\Gamma \int_\Omega p^2dV\quad\text{provided that }q\cdot n= 0\quad\text{on }\partial\Omega. \] The optimal constants \(\Gamma\) and \(\widetilde\Gamma\) are related by \[ \Gamma= C-1\quad\text{and }\widetilde\Gamma= \widetilde C- 1 \] to the optimal constants \(C\) and \(\widetilde C\) in the inequalities \[ D(w)\leq Cd(w)\quad\text{for all }w\in W\oplus \text{Ker rot} \] and \[ D(w)\leq \widetilde Cr(w)\quad\text{for all }w\in W\oplus \text{Ker div}, \] where \[ D(w,\widetilde w)= d(w,\widetilde w)+ r(w,\widetilde w), \] \[ d(w,\widetilde w)= \int_\Omega \text{div }w\text{ div }\widetilde w dV\quad\text{and}\quad r(w,\widetilde w)= \int_\Omega \text{rot }w\cdot\text{rot }\widetilde w dV, \] and \(W\) is the subspace in the decomposition \[ H^1_0(\Omega)^3= \text{Ker div}\oplus \text{Ker rot}\oplus W. \] It is also shown how these optimal constants \(\Gamma\), \(\widetilde\Gamma\), \(C\) and \(\widetilde C\) are connected with the ones in related eigenvalue problems and Stokes equations.
scientific article; zbMATH DE number 1270165