Doubling measure and regularity to <i>K</i> -quasiminimizers of double-phase energy

Abstract We consider an integral functional involving the double-phase operator in a metric measure space equipped with a doubling measure. On the basis of suitable hypotheses on the function governing the phase changes, we design a unifying approach to establish Sobolev-Poincaré inequalities. By using such inequalities together with a Caccioppoli type estimate, we obtain the local boundedness and the local Hölder continuity of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K -quasiminimizers of the double-phase energy. Here, we provide a direct approach for establishing the Sobolev-Poincaré inequalities and local Hölder continuity, that is, we do not use neither the method of separation of phases nor auxiliary frozen functional. Finally, we establish the local gradient higher integrability for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K -quasiminimizers of the integral functional. We prove our results via the De Giorgi’s method, imposing that the involved measure is just doubling.