$mathcal{B}_{1}$ classes of DeGiorgi-Ladyzhenskaya-Uraltseva 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 Uraltseva 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 $(p, q)$-phase growth including the ,,singular-degenerate parabolic case $p<2<q$.