arXiv subjects
Publications and source records attributed to Wojciech Ozanski.
We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure $p$ appearing in the stationary Stokes equations is the Lagrange multiplier of the constraint $\mathrm{div}\, u =0$.
We extend the generalised comparison principle for the Monge-Ampère equation due to Rauch & Taylor (Rocky Mountain J. Math. 7, 1977) to nonconvex domains. From the generalised comparison principle we deduce bounds (from above and below) on solutions of the Monge-Ampère equation with sign-changing right-hand side. As a consequence, if the right-hand side is nonpositive (and does not vanish almost everywhere) then the equation equipped with constant boundary condition has no solutions. In particular, due to a connection between the two-dimensional Navier-Stokes equations and the Monge-Ampère equation, the pressure $p$ in 2D Navier-Stokes equations on a bounded domain cannot satisfy $Δp \leq 0$ in $Ω$ unless $Δp \equiv 0$ (at any fixed time). As a result at any time $t>0$ there exists $z\in Ω$ such that $Δp (z,t) =0$.