Partial regularity for parabolic systems of double phase type
Jihoon Ok, Giovanni Scilla, Bianca Stroffolini
TL;DR
This work addresses partial regularity for parabolic systems with generalized double-phase growth $H(z,s)=s^p+a(z)s^q$, under $\tfrac{2n}{n+2}<p\le q$ and $a\in C^{0,α,α/2}$. The authors develop a unified framework based on generalized Orlicz growth, mollification-based Caccioppoli inequalities, and $\mathcal{A}$-caloric approximation to obtain linearization and excess-decay estimates. A decay mechanism for the excess functional yields a standard regularity bootstrap, proving local Hölder continuity of the spatial gradient $D\mathbf{u}$ on a full-measure open set and a measure-zero singular set. The results extend parabolic regularity theory to nonuniform (double-phase) growth in a nondegenerate setting and provide a foundation for future work on degenerate cases, with a cohesive method that treats both $p$- and $(p,q)$-phases uniformly.
Abstract
We study partial regularity for nondegenerate parabolic systems of double phase type, where the growth function is given by $H(z,s)=s^p+a(z)s^q$, $z=(x,t)\inΩ_T$, with $\tfrac{2n}{n+2}<p\le q$ and $a(z)$ a nonnegative $C^{0,α,\fracα{2}}$-continuous function for some $α\in(0,1]$. As the main result we prove that if $q< \min \{p+\tfrac{αp }{n+2}, p+1 \}$ the spatial gradient of any weak solution is locally Hölder continuous, except on a set of measure zero.