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We prove some new Hardy-type inequalities for the Jacobi weight function. The resulting inequalities contain additional terms with the weight functions characteristic of Poincaré–Friedrichs inequalities. One of the constants in the inequality is unimprovable. We apply the inequalities to extending the available classes of univalent analytic functions in simply-connected domains and find univalence conditions in terms of estimates for the Schwartz derivative of an analytic function on the unit disk, the exterior of the unit disk, and the right half-plane. [ABSTRACT FROM AUTHOR]

We derive bounds for the constants in Poincaré–Friedrichs inequalities with respect to mesh-dependent norms for complexes of discrete distributional differential forms. A key tool is a generalized flux reconstruction which is of independent interest. The results apply to piecewise polynomial de Rham sequences on bounded domains with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

The Schlögl system is governed by a nonlinear reaction–diffusion partial differential equation with a cubic nonlinearity that determines three constant equilibrium states. It is a classical example of a chemical reaction system that is bistable. The constant equilibrium that is enclosed by the other two constant equilibrium points is unstable. In this paper, Robin boundary feedback laws are presented that stabilize the system in a given stationary state or more generally in a given time-dependent desired system orbit. The exponential stability of the closed loop system with respect to the L 2 -norm is proved. In particular, it is shown that with the boundary feedback law the unstable constant equilibrium point can be stabilized. [ABSTRACT FROM AUTHOR]