$\mathcal{B}_{1}$ classes of DeGiorgi-Ladyzhenskaya-Ural'tseva and their applications to elliptic and parabolic equations with generalized Orlicz growth conditions

TL;DR

This paper introduces generalized $\

Contribution

It extends classical DeGiorgi-Ladyzhenskaya-Ural'tseva classes to new elliptic and parabolic $\

Findings

Proves pointwise continuity of solutions under new growth conditions

Includes cases with variable exponent and $(p, q)$-phase growth

Addresses singular-degenerate parabolic equations

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 $p<2<q$.