${\mathcal{B}}_1$ classes of DeGiorgi-Ladyzhenskaya-Ural'tseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions

We introduce elliptic and parabolic ${\mathcal{B}}_1$ classes that generalize the well-known ${\mathfrak{B}}_p$ classes of DeGiorgi, Ladyzhenskaya and Ural’tseva with $p>1$. New classes are applied to prove pointwise continuity of solutions of elliptic and parabolic equations with nonstandard growth conditions. Our considerations cover new cases of variable exponent and $(p, q)$-phasegrowthincludingthe,,singular-degenerat{e}^{\mathrm{\prime }\mathrm{\prime }}paraboliccase$p<2<q$.