Regularity for minimizers for functionals of double phase with variable exponents

Abstract

Abstract The functionals of double phase type $$\begin{array}{} \displaystyle {\cal H} (u):= \int \left(|Du|^{p} + a(x)|Du|^{q} \right) dx, ~~ ~~~(q \gt p \gt 1,~~a(x)\geq 0) \end{array}$$ 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.