Regularity for minimizers for functionals of double phase with variable exponents

The functionals of double phase type array H (u):= ∫ (|Du|ᵖ + a(x)|Du|q ) dx, ~~ ~~~(q > p > 1,~~a(x)≥ 0) array are introduced in the epoch-making paper by Colombo-Mingione [1] for constants p and q , and investigated by them and Baroni. They obtained sharp regularity results for minimizers of such functionals. In this paper we treat the case that the exponents are functions of x and partly generalize their regularity results.