In this talk we will present regularity results for a class of double phase integrals characterized by nonstandard growth conditions, which constitute an important sub-field of the calculus of variations. The focus is on regularity theory, specifically local and global higher integrability, for quasiminimizers of a double phase integral characterized by the so-called $(p,q)$-growth. The proofs are based purely on variational methods, in the general context of metric measure spaces with a doubling measure and a Poincaré inequality. The main novelty is the development of an intrinsic approach to Sobolev--Poincaré inequalities which takes into account the double phase geometry of the integral.

This is a joint work with Juha Kinnunen (Aalto University) and Cintia Pacchiano Camacho (Calgary University).