Boundary regularity for quasiminima of double-phase problems on metric spaces

We use variational methods to prove pointwise estimates near a boundary point for quasiminima of integrals of p, q -Laplace type, whose integrand switches between p and q growth rate. This study is conducted in the general setting of metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality, the obtained results are new even in the Euclidean case. The proofs rely on a careful phase analysis and estimates in the intrinsic geometries and they are based on De Giorgi method to obtain a sufficient condition for Hölder continuity at a boundary point for ( p, q )-quasiminimizers.