Minima of quasisuperminimizers

TL;DR

This paper investigates the behavior of quasisuperminimizers under taking minima, improves bounds on their quasisuperminimizing constants, and provides new examples where the combined constant exceeds the individual ones.

Contribution

It refines the known upper bounds for the quasisuperminimizing constant of the minimum of two such functions and introduces the first examples with a larger constant Q than Q_2.

Findings

Improved upper bounds for the quasisuperminimizing constant Q.

First examples showing Q > Q_2.

Analysis of the blowup of constants in pasting lemmas.

Abstract

Let u_i be a Q_i-quasisuperminimizer, i=1,2, and u=min{u_1,u_2}, where 1 <= Q_1 <= Q_2. Then u is a quasisuperminimizer, and we improve upon the known upper bound (due to Kinnunen and Martio) for the optimal quasisuperminimizing constant Q of u. We give the first examples with Q>Q_2, and show that in general Q>Q_2 whenever Q_1 >1. We also study the blowup of the quasisuperminimizing constant in pasting lemmas.