Regularity for Double Phase Functionals with Two Modulating Coefficients

In this paper, we establish regularity results for local minimizers of functionals with non-standard growth conditions and non-uniform ellipticity properties. The model case is given by the double phase integral with two modulating coefficients \(\begin{aligned} w\mapsto \int [a(x)|Dw|^p+b(x)|Dw|^q] dx, \qquad 1<p<q, \qquad a(\cdot ),b(\cdot )\ge 0, \end{aligned}\) $$\begin{array}{c}\hfill w\mapsto \int [a(x)|Dw{|}^p+b(x)|Dw{|}^q]dx,1<p<q,a(·),b(·)\ge 0,\end{array}$$ with \(0<\mu \le a(\cdot )+b(\cdot )\) $0<\mu \le a(·)+b(·)$ . Here, the coefficient \(b(\cdot )\) $b(·)$ is assumed to be Hölder continuous and the coefficient \(a(\cdot )\) $a(·)$ is assumed to be uniformly continuous.