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

Abstract

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 $(p,q)$-phase growth including the ,,singular-degenerate'' parabolic case $p<2<q$.