On monotone extensions of boundary data

Summary

A functionf ∈C (Ω),$\Omega \subseteq \mathbb{R}^s$ is called monotone on Ω if for anyx, y ∈ Ω the relation x − y ∈ ∝ ^s₊ impliesf(x)≧f(y). Given a domain$\Omega \subseteq \mathbb{R}^s$ with a continuous boundary ∂Ω and given any monotone functionf on ∂Ω we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of Ω and agree withf on ∂Ω. In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when Ω is the unit square [0, 1]² in ∝², strictly monotone analytic boundary data admit a monotone analytic extension.