Minima of quasisuperminimizers

We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for QQ-quasisuperminimizers. More precisely, if uiui is a QiQi-quasisuperminimizer, i=1,2i=1,2, where 1<Q1≤Q21<Q1≤Q2, then u=min{u1,u2}u=min{u1,u2} is a QQ-quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant QQ. We provide the first examples of this phenomenon, i.e. that Q>Q2Q>Q2.

In addition to lower bounds for the optimal quasisuperminimizing constant of uu we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q=Q1Q2Q=Q1Q2 turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers.