Partial regularity for degenerate parabolic systems with non-standard growth and discontinuous coefficients

TL;DR

This paper proves partial regularity results for weak solutions to certain degenerate parabolic systems with non-standard growth, extending regularity theory to systems with discontinuous coefficients and variable exponents.

Contribution

It establishes partial Hölder continuity for solutions to degenerate parabolic systems with variable exponent p(z) and discontinuous coefficients under specific conditions.

Findings

Solutions are locally Hölder continuous outside a measure zero set.

Regularity holds under VMO-type conditions on coefficients.

Results extend partial regularity theory to systems with non-standard growth.

Abstract

This article studies the partial H\"older continuity of weak solutions to certain degenerate parabolic systems whose model is the differentiable parabolic $p(x,t)$-Laplacian system, \begin{equation*}\partial_t u-\operatorname{div}[\mu(z)(1+|Du|^2)^{\frac{p(z)-2}{2}}Du]=0,\qquad p(z)\geq2.\end{equation*} Here, the exponential function $p(z)$ satisfies a logarithmic continuity condition. We show that if $\mu(z)$ satisfies a certain VMO-type condition, then $u$ is locally H\"older continuous except for a measure zero set.