Higher integrability and stability of $(p,q)$-quasiminimizers

TL;DR

This paper proves higher integrability and stability results for quasiminimizers of a $(p,q)$-Dirichlet integral in metric measure spaces, using variational methods and assuming improved Newtonian space conditions.

Contribution

It introduces new higher integrability and stability results for $(p,q)$-quasiminimizers in metric measure spaces, expanding understanding of their regularity and dependence on exponents.

Findings

Higher integrability of upper gradients of quasiminimizers

Stability of quasiminimizers with respect to exponents $p$ and $q$

Results applicable in doubling metric measure spaces with Poincaré inequality

Abstract

Using purely variational methods, we prove local and global higher integrability results for upper gradients of quasiminimizers of a $(p,q)$-Dirichlet integral with fixed boundary data, assuming it belongs to a slightly better Newtonian space. We also obtain a stability property with respect to the varying exponents $p$ and $q$. The setting is a doubling metric measure space supporting a Poincar\'e inequality.